Indices, Exponentials and Logarithms Study Guide

Review of Index Laws

• m,//// Index Notation and Base Concepts

  • Index notation represents repeated multiplication of the same factor.

  • General form: $a^n$, where $a$ is the base and $n$ is the index (exponent or power).

  • Examples of repeated multiplication:

    • $2 \times 2 \times 2 = 2^3$

    • $5 \times x \times x \times x \times x = 5x^4$

  • Single terms without explicit powers have an index of $1$:

    • $a = a^1$

    • $5a = 5^1 \times a^1$

• Fundamental Index Laws

  • Index Law for Multiplication:

    • Formula: $a^m \times a^n = a^{m+n}$

    • Rule: Retain the common base and add the indices.

  • Index Law for Division:

    • Formula: $a^m \frac{a^m}{a^n} = a^{m-n}$ or $a^m \frac{a^m}{a^n} = a^{m-n}$

    • Rule: Retain the common base and subtract the indices.

  • Index Law for Power of a Power:

    • Formula: $(a^m)^n = a^{m \times n}$

    • Rule: Retain the base and multiply the indices.

  • Index Law for Brackets (Product Power Law):

    • Formula: $(a \times b)^m = a^m \times b^m$

    • Rule: Distribute the index number across each factor in the base.

  • Index Law for Fractions (Quotient Power Law):

    • Formula: $\\left(\frac{a}{b}\\right)^m = \frac{a^m}{b^m}$

    • Rule: Distribute the index number across both numerator and denominator bases.

  • The Zero Index:

    • Formula: $a^0 = 1$ (where $a \neq 0$)

    • Rule: Any non-zero number raised to the power of zero equals $1$.

• Practical Applications of Index Powers

  • Powers of $2$ are utilized to measure digital data sizes (bytes, kilobytes, megabytes) and model bacterial population growth.

  • Powers of $10$ are utilized to calculate earthquake intensity scales (Richter scale) and acoustic sound levels (decibels).

• Comprehensive Worked Examples

  • Worked Example 1: Multiplication and Division

    • Simplify $x^5 \times x^4$:

      • Add indices for base $x$: $x^{5+4} = x^9$.

    • Simplify $3a^2b \times 4ab^3$:

      • Multiply numerical coefficients ($3 \times 4 = 12$) and add indices for bases $a$ and $b$ ($a^2 \times a^1 = a^3$, $b^1 \times b^3 = b^4$): $12a^3b^4$.

    • Simplify $m^7 \div m^5$:

      • Subtract indices for common base $m$: $m^{7-5} = m^2$.

    • Simplify $4x^2y^5 \div (8xy^2)$:

      • Express as a fraction: $\frac{4x^2y^5}{8xy^2}$.

      • Divide coefficients ($\frac{4}{8} = \frac{1}{2}$) and subtract indices ($x^{2-1} = x^1$, $y^{5-2} = y^3$): $\frac{xy^3}{2} = \frac{1}{2}xy^3$.

  • Worked Example 2: Brackets and Powers

    • Simplify $(a^3)^4$:

      • Multiply indices: $a^{3 \times 4} = a^{12}$.

    • Simplify $(2y^5)^3$:

      • Distribute power $3$ to base $2$ and base $y$: $2^3 \times y^{5 \times 3} = 8y^{15}$.

    • Simplify $\left(\frac{a^2}{2b}\right)^3$:

      • Distribute index $3$ to numerator and denominator bases: $\frac{(a^2)^3}{2^3 \times b^3} = \frac{a^6}{8b^3}$.

  • Worked Example 3: Zero Index Expressions

    • Evaluate $4a^0$:

      • $4a^0 = 4 \times 1 = 4$.

    • Evaluate $2p^0 + (3p)^0$:

      • $2p^0 + (3p)^0 = 2(1) + 1 = 2 + 1 = 3$.

      • Note: $(3p)^0 = 1$ whereas $3p^0 = 3 \times 1 = 3$.

  • Worked Example 4: Combining Multiple Index Laws

    • Simplify $3x^2y^3 \times 6xy^4 \div (2y^2)$:

      • Step 1 (Multiplication): $(3 \times 6) \times (x^2 \times x^1) \times (y^3 \times y^4) = 18x^3y^7$.

      • Step 2 (Division): $\frac{18x^3y^7}{2y^2} = (18 \div 2)x^3y^{7-2} = 9x^3y^5$.

    • Simplify $\frac{3(xy^2)^3 \times 4x^4y^2}{8x^2y}$:

      • Step 1 (Remove brackets): $3(x^3y^6) \times 4x^4y^2 = 3x^3y^6 \times 4x^4y^2$.

      • Step 2 (Combine numerator): $(3 \times 4) \times (x^3 \times x^4) \times (y^6 \times y^2) = 12x^7y^8$.

      • Step 3 (Divide denominator): $\frac{12x^7y^8}{8x^2y} = \frac{12}{8} x^{7-2} y^{8-1} = \frac{3x^5y^7}{2}$.

Negative Indices

• Mathematical Derivation of Negative Indices

  • Combining the index law for division with the zero power identity ($a^0 = 1$):

    • $a^0 \div a^n = a^{0-n} = a^{-n}$

    • $a^0 \div a^n = 1 \div a^n = \frac{1}{a^n}$

    • Equating both expressions yields: $a^{-n} = \frac{1}{a^n}$

  • Reciprocal relationship in denominators:

    • $\frac{1}{a^{-n}} = 1 \div a^{-n} = 1 \div \frac{1}{a^n} = 1 \times a^n = a^n$

    • Equating expressions yields: $\frac{1}{a^{-n}} = a^n$

• Key Negative Index Laws and Reciprocals

  • Standard form: $a^{-m} = \frac{1}{a^m}$

    • Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

  • Denominator form: $\frac{1}{a^{-m}} = a^m$

    • Example: $\frac{1}{2^{-3}} = 2^3 = 8$

  • Reciprocal identities:

    • $a^{-1} = \frac{1}{a}$ (the multiplicative inverse/reciprocal of $a$)

    • $\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$

    • $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}$

• Applications of Negative Powers of 2

  • Radioactive half-life calculations (time taken for a substance to halve in quantity).

  • Nuclear waste management and diagnostic nuclear medicine dosage computations.

• Comprehensive Worked Examples

  • Worked Example 5: Expressions with Positive Indices

    • Express $b^{-4}$ using positive indices: $b^{-4} = \frac{1}{b^4}$.

    • Express $3x^{-4}y^2$ using positive indices:

      • Identify base with negative exponent: base $x$ has power $-4$.

      • Rewrite: $\frac{3}{1} \times \frac{1}{x^4} \times \frac{y^2}{1} = \frac{3y^2}{x^4}$.

    • Express $\frac{5}{x^{-3}}$ using positive indices:

      • $5 \times \frac{1}{x^{-3}} = 5 \times x^3 = 5x^3$.

  • Worked Example 6: Simplifying Expressions with Negative Indices

    • Simplify $\frac{2a^3b^2}{a^5b^3}$ with positive indices:

      • Apply division law: $2 \times a^{3-5} \times b^{2-3} = 2a^{-2}b^{-1}$.

      • Convert to positive indices: $2 \times \frac{1}{a^2} \times \frac{1}{b} = \frac{2}{a^2b}$.

    • Simplify $\frac{4m^{-2}n^3}{8m^5n^{-4}}$ with positive indices:

      • Simplify numerical coefficients: $\frac{4}{8} = \frac{1}{2}$.

      • Subtract indices: $m^{-2-5} = m^{-7}$, $n^{3-(-4)} = n^{3+4} = n^7$.

      • Combine: $\frac{1}{2} \times m^{-7} \times n^7 = \frac{n^7}{2m^7}$.

  • Worked Example 7: Complex Multi-Step Expressions

    • Simplify $(x^3y^{-1})^3 \times (x^{-2}y^3)^{-5}$:

      • Expand brackets: $(x^{3 \times 3}y^{-1 \times 3}) \times (x^{-2 \times (-5)}y^{3 \times (-5)}) = (x^9y^{-3}) \times (x^{10}y^{-15})$.

      • Add indices of common bases: $x^{9+10}y^{-3+(-15)} = x^{19}y^{-18}$.

      • Convert to positive indices: $\frac{x^{19}}{y^{18}}$.

    • Simplify $\frac{(p^{-2}q)^4}{5pq^{-6}} \div \left(\frac{p^{-2}}{q^3}\right)^{-3}$:

      • Expand power terms: $\frac{p^{-8}q^4}{5pq^{-6}} \div \frac{p^6}{q^{-9}}$.

      • Convert division to multiplication by multiplying by the reciprocal: $\frac{p^{-8}q^4}{5pq^{-6}} \times \frac{q^{-9}}{p^6}$.

      • Multiply numerators and denominators: $\frac{p^{-8}q^{4+(-9)}}{5p^{1+6}q^{-6}} = \frac{p^{-8}q^{-5}}{5p^7q^{-6}}$.

      • Apply division rule for powers: $\frac{p^{-8-7}q^{-5-(-6)}}{5} = \frac{p^{-15}q^1}{5} = \frac{q}{5p^{15}}$.

Scientific Notation

• Standard Definition of Scientific Notation

  • Scientific notation expresses numbers in the standard form:

    • $a \times 10^m$

    • Where $1 \le a < 10$ (for positive numbers) or $-10 < a \le -1$ (for negative numbers), and $m$ is an integer ($m \in \mathbb{Z}$).

  • Large numbers use positive powers of $10$:

    • $24\,800\,000 = 2.48 \times 10^7$

    • $9\,020\,000\,000 = 9.02 \times 10^9$

  • Small numbers use negative powers of $10$:

    • $0.00307 = 3.07 \times 10^{-3}$

    • $-0.0000012 = -1.2 \times 10^{-6}$

• Significant Figures (s.f.) and Rounding Rules

  • Significant figures are counted from left to right starting at the first non-zero digit.

  • In scientific notation, the single digit to the left of the decimal point represents the first significant figure.

  • Examples:

    • $20\,190\,000 = 2.019 \times 10^7$ (expressed to $4$ significant figures).

    • $0.0042031 = 4.20 \times 10^{-3}$ (expressed to $3$ significant figures; trailing zero is retained to preserve accuracy).

• Calculator Input Notation

  • Scientific notation is entered using key functions such as x10^x, EE, or Exp.

  • Example display readout: 2.3E-4 represents $2.3 \times 10^{-4}$.

• Mass and Physical Constants Examples

  • Distance from Earth to Sun: $150\text{ million km} = 1.5 \times 10^8\,km$.

  • Mass of a neutron: $0.000000000000000000000000001675\,kg = 1.675 \times 10^{-27}\,kg$.

  • Mass of a bacterial cell: $0.00000000000095\,g = 9.5 \times 10^{-13}\,g$.

  • Einstein's Mass-Energy Equivalence Formula: $E = mc^2$

    • Where $E$ is energy in Joules ($J$), $m$ is mass in kilograms ($kg$), and $c \approx 3 \times 10^8\,m/s$ (speed of light).

• Comprehensive Worked Examples

  • Worked Example 8: Converting to Basic Numerals

    • Convert $5.016 \times 10^5$ to a basic numeral:

      • Shift decimal point $5$ places to the right: $501\,600$.

    • Convert $3.2 \times 10^{-7}$ to a basic numeral:

      • Shift decimal point $7$ places to the left: $0.00000032$.

  • Worked Example 9: Converting to Scientific Notation with Rounding

    • Write $5\,218\,300$ in scientific notation to $3$ significant figures:

      • First non-zero digit is $5$. Place decimal point after it: $5.2183 \times 10^6$

      • Round to $3$ s.f. (fourth digit is $8 \ge 5$, so round up): 5.22 \times 10^6$.\n * Write 0.0042031$in scientific notation to$3 significant figures:\n * First non-zero digit is 4$at third decimal place:$4.2031 \times 10^{-3}\n * Round to 3$s.f. (fourth digit is$3 < 5$, round down):$4.20 \times 10^{-3}.\n * **Worked Example 10: Calculator Computations**\n * Evaluate \frac{\sqrt{2.61 \times 10^4}}{3.2 \times 10^{-2}}$in scientific notation to$3 s.f.:\n * Calculator output: 5048.598…\n * Scientific notation to 3$s.f.:$5.05 \times 10^3$.

Fractional Indices

• Rational Indices and Radicals

  • A rational index is an index expressed as a fraction ($\frac{m}{n}$).

  • Fundamental definition of unit fractions as roots:

    • $a^{\frac{1}{n}} = \sqrt[n]{a}$

    • $\sqrt[n]{a}$ denotes the $n$th root of $a$.

  • Equivalence proof via index laws:

    • $\sqrt{9} \times \sqrt{9} = 3 \times 3 = 9$

    • $9^{\frac{1}{2}} \times 9^{\frac{1}{2}} = 9^{\frac{1}{2} + \frac{1}{2}} = 9^1 = 9$

    • Therefore, $\sqrt{9} = 9^{\frac{1}{2}}$.

    • $\sqrt[3]{8} \times \sqrt[3]{8} \times \sqrt[3]{8} = 2 \times 2 \times 2 = 8$

    • $8^{\frac{1}{3}} \times 8^{\frac{1}{3}} \times 8^{\frac{1}{3}} = 8^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}} = 8^1 = 8$

    • Therefore, $\sqrt[3]{8} = 8^{\frac{1}{3}}$.

• General Fractional Index Formulae

  • Positive fractional indices:

    • $a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m = (\sqrt[n]{a})^m$

    • $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}$

  • Negative fractional indices:

    • $a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{(\sqrt[n]{a})^m}$

• Comprehensive Worked Examples

  • Worked Example 11: Converting Roots to Index Form

    • Express $\sqrt{15}$ in index form: $15^{\frac{1}{2}}$.

    • Express $\sqrt{7x^5}$ in index form:

      • $(7x^5)^{\frac{1}{2}} = 7^{\frac{1}{2}}x^{5 \times \frac{1}{2}} = 7^{\frac{1}{2}}x^{\frac{5}{2}}$.

    • Express $3\sqrt[4]{x^7}$ in index form:

      • $3(x^7)^{\frac{1}{4}} = 3x^{\frac{7}{4}}$.

    • Express $10\sqrt{10}$ in index form:

      • $10^1 \times 10^{\frac{1}{2}} = 10^{1 + \frac{1}{2}} = 10^{\frac{3}{2}}$.

  • Worked Example 12: Converting Fractional Indices to Surd Form

    • Express $3^{\frac{1}{5}}$ in surd form: $\sqrt[5]{3}$.

    • Express $5^{\frac{2}{3}}$ in surd form:

      • Method 1: $(5^2)^{\frac{1}{3}} = \sqrt[3]{25}$.

      • Method 2: $\left(5^{\frac{1}{3}}\right)^2 = (\sqrt[3]{5})^2$.

  • Worked Example 13: Non-Calculator Evaluations

    • Evaluate $16^{\frac{1}{2}}$: $\sqrt{16} = 4$.

    • Evaluate $27^{-\frac{1}{3}}$:

      • $\frac{1}{27^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3}$.

    • Evaluate $16^{\frac{3}{4}}$:

      • $\left(16^{\frac{1}{4}}\right)^3 = (\sqrt[4]{16})^3 = 2^3 = 8$

      • (Alternative method: $\sqrt[4]{16^3} = \sqrt[4]{4096} = 8$, root-first is simpler).

Exponential Equations

• Definition and Core Algebraic Property

  • A simple exponential equation has the form $a^x = b$, where $a > 0$, $b > 0$, and $a \neq 1$

  • Equating Bases Rule: If $a^x = a^y$, then $x = y$.

  • Exponential equations contain pronumerals within the index positions.

• Comprehensive Worked Examples

  • Worked Example 14: Solving Simple Equations

    • Solve $2^x = 16$:

      • Express $16$ as a power of base $2$: $2^x = 2^4$.

      • Equate powers: $x = 4$

    • Solve $3^x = \frac{1}{9}$:

      • Express $\frac{1}{9}$ with base $3$: $\frac{1}{9} = \frac{1}{3^2} = 3^{-2}$.

      • Equate powers: $3^x = 3^{-2} \implies x = -2$.

    • Solve $25^x = 125$:

      • Express both sides using lowest common base $5$:

      • $(5^2)^x = 5^3 \implies 5^{2x} = 5^3$

      • Equate powers: $2x = 3 \implies x = \frac{3}{2}$.

  • Worked Example 15: Variable on Both Sides

    • Solve $3^{2x-1} = 27^x$:

      • Convert $27$ to base $3$: $27^x = (3^3)^x = 3^{3x}$.

      • Set expressions equal: $3^{2x-1} = 3^{3x}$.

      • Equate indices: $2x - 1 = 3x$.

      • Solve for $x$: $-1 = 3x - 2x \implies x = -1$.

Exponential Functions and Their Graphs

• Key Characteristics of Exponential Functions

  • Standard equation: $y = a^x$ (where $a > 0$ and $a \neq 1$).

  • $y$-intercept: Always located at $(0, 1)$ because $a^0 = 1$.

  • Asymptote: A straight line that a curve approaches continuously without ever touching.

    • For $y = a^x$, the horizontal asymptote is the $x$-axis, given by equation $y = 0$

  • Domain: All real $x$ ($x \in \mathbb{R}$).

  • Range: $y > 0$

• Reflections of Exponential Curves

  • Reflection in the $x$-axis: $y = -a^x$ (note: $y = -1 \times a^x$). Intercept at $(0, -1)$.

  • Reflection in the $y$-axis: $y = a^{-x} = \left(\frac{1}{a}\right)^x$. Intercept at $(0, 1)$.

• Graphical & Algebraic Intersections

  • The intersection point of $y = a^x$ and horizontal line $y = c$ is found by solving $a^x = c$.

• Comprehensive Worked Examples

  • Worked Example 16: Comparative Graph Sketching

    • Sketch $y = 2^x$, $y = 3^x$, $y = 4^x$ on identical axes:

      • All curves pass through $(0, 1)$.

      • For $x > 0$, higher base values yield steeper growth: points at $x = 1$ are $(1, 2)$, $(1, 3)$, and $(1, 4)$.

      • As $x \to -\infty$, all curves approach the asymptote $y = 0$.

  • Worked Example 17: Sketching Reflections

    • Sketch $y = 3^x$, $y = -3^x$, and $y = 3^{-x}$:

      • $y = 3^x$: Passes through $(0, 1)$ and $(1, 3)$.

      • $y = -3^x$: Reflected across $x$-axis. Passes through $(0, -1)$ and $(1, -3)$.

      • $y = 3^{-x}$: Reflected across $y$-axis. Passes through $(0, 1)$, $(-1, 3)$, and $\left(1, \frac{1}{3}\right)$.

  • Worked Example 18: Solving Graphically

    • Find intersection of $y = 2^x$ and $y = 8$:

      • Set $2^x = 8 \implies 2^x = 2^3 \implies x = 3$.

      • Point of intersection coordinates: $(3, 8)$.

Exponential Growth and Decay

• Model Formulations

  • Standard exponential equation: $A = k a^t$ or $A = A_0 \left(1 \pm \frac{r}{100}\right)^n$

    • $A$: final amount / value

    • $A_0$ (or $k$): initial quantity (subscript $0$ denotes time $t = 0$)

    • $r$: percentage rate per time period

    • $n$ (or $t$): number of time periods

  • Exponential Growth: Occurs when base $a > 1$

    • Base calculation: $a = 1 + \frac{r}{100}$

  • Exponential Decay: Occurs when base $0 < a < 1$

    • Base calculation: $a = 1 - \frac{r}{100}$

• Comprehensive Worked Examples

  • Worked Example 19: Rule Construction

    • Painting valued at $\$100\,000$ increasing by $14\%$ p.a.:

      • $A_0 = 100\,000$, $r = 14$, growth ($+$).

      • $A = 100\,000 \left(1 + \frac{14}{100}\right)^n \implies A = 100\,000(1.14)^n$

    • City population of $50\,000$ decreasing by $12\%$ p.a.:

      • $P_0 = 50\,000$, $r = 12$, decay ($-$).

      • $P = 50\,000 \left(1 - \frac{12}{100}\right)^n \implies P = 50\,000(0.88)^n$

  • Worked Example 20: Applying Rules and Trial and Error

    • Flat valued at $\$600\,000$ growing at $9\%$ p.a.:

      • Rule: $V = 600\,000(1.09)^n$

      • Value in $1$ year ($n = 1$): $V = 600\,000(1.09)^1 = \$654\,000$

      • Value in $3$ years ($n = 3$): $V = 600\,000(1.09)^3 = \$777\,017.40$

      • Time to reach $\$900\,000$:

        • Set $600\,000(1.09)^n = 900\,000 \implies (1.09)^n = 1.5$

        • Trial $n = 4$: $1.09^4 \approx 1.4116 \implies V = \$846\,949$

        • Trial $n = 5$: $1.09^5 \approx 1.5386 \implies V = \$923\,174$

        • Trial $n = 4.7$: $1.09^{4.7} \approx 1.49936 \implies V \approx \$899\,613$

        • Conclusion: Approximately $4.7$ years.

Introduction to Logarithms

• Historical and Conceptual Foundation

  • Logarithms ("logical arithmetic") were invented by John Napier in the 17th century to simplify complex arithmetic.

  • A logarithm is the power to which a base must be raised to yield a given number.

• Fundamental Equivalent Relationship

  • $\log_a(y) = x \iff a^x = y$

  • Conditions: base $a > 0$, argument $y > 0$.

  • Read as: "The logarithm of $y$ to the base $a$ is equal to $x$".

• Comprehensive Worked Examples

  • Worked Example 21: Converting Between Forms

    • Write equivalent index statement for $\log_{10}(1000) = 3$: $10^3 = 1000$.

    • Write equivalent logarithmic statement for $2^5 = 32$: $\log_2(32) = 5$.

  • Worked Example 22: Logarithm Evaluation

    • Evaluate $\log_2(8)$: Solve $2^x = 8 \implies x = 3$.

    • Evaluate $\log_5(625)$: Solve 5^x = 625 \implies x = 4$.\n * Evaluate \log_3\left(\frac{1}{9}\right)$: Solve$3^x = 3^{-2} \implies x = -2$.

    • Evaluate $\log_{10}(0.001)$: Solve 10^x = 10^{-3} \implies x = -3$.\n * Calculator evaluation to 3 d.p.:\n * \log_{10}(7) \approx 0.845\n * \log_{10}(0.5) \approx -0.301\n * **Worked Example 23: Solving Logarithmic Equations**\n * Solve \log_4(64) = x:\n * Convert to index form: 4^x = 64 \implies 4^x = 4^3 \implies x = 3$.

    • Solve $\log_2(x) = 6$:

      • Convert to index form: x = 2^6 = 64$.\n\n# Logarithmic Scales\n\n* **Definition and Order of Magnitude**\n * Logarithmic scales compress wide ranges of data values for effective visualization.\n * **Order of Magnitude**: The power of 10 used when writing a number in scientific notation.\n * Increasing a number by n$orders of magnitude multiplies it by$10^n$.

    • Decreasing a number by $n$ orders of magnitude divides it by 10^n$.\n * Domains using logarithmic scales:\n * Seismology (Richter scale / moment magnitude scale)\n * Acoustics (Decibels dB)\n * Chemistry (pH level of acidity)\n * Astronomy (Stellar brightness / magnitudes)\n * Finance and Population metrics.\n\n* **Linearization of Exponential Functions**\n * For exponential model A = A_0(1 + r)^t\n * Taking \log_{10}$yields:$\log_{10}(A) = [\log_{10}(1+r)] \cdot t + \log_{10}(A_0)\n * This matches straight-line form y = mt + c$, where slope$m = \log_{10}(1+r)$and vertical intercept$c = \log_{10}(A_0).\n\n* **Comprehensive Worked Examples**\n * **Worked Example 24: Interpreting Logarithmic Magnitude Charts**\n * Earthquake data: Papua New Guinea (6$), Fiji ($5$), Alaska ($4$), Puerto Rico ($2).\n * Magnitude difference PNG (6$) vs Alaska ($4$):$6 - 4 = 2\n * PNG earthquake is 10^2 = 100 times more powerful than Alaska's.\n * Magnitude difference Fiji (5$) vs Puerto Rico ($2$):$5 - 2 = 3\n * Fiji earthquake is 10^3 = 1000 times more powerful than Puerto Rico's.\n * **Worked Example 25: Constructing Logarithmic Data Tables**\n * Bacteria population data P$over time$t:\n * t = [0, 1, 2, 3, 4, 5]\n * P = [20, 200, 2\,000, 20\,000, 200\,000, 2\,000\,000]\n * Compute \log_{10}(P)$:$[1.3, 2.3, 3.3, 4.3, 5.3, 6.3].\n * Graph of \log_{10}(P)$versus$t$forms a straight line, confirming an exponential relationship between$P$and$t\n\n# Laws of Logarithms\n\n\n\n* **Derivation of Logarithm Laws**\n * **Logarithm Law for Addition (Product Rule)**:\n * Formula: \log_a(x) + \log_a(y) = \log_a(xy)\n * Proof: Let x = a^m \implies m = \log_a(x)$and$y = a^n \implies n = \log_a(y).\n * Multiply: xy = a^m \times a^n = a^{m+n}.\n * Convert to log form: \log_a(xy) = m + n = \log_a(x) + \log_a(y).\n * **Logarithm Law for Subtraction (Quotient Rule)**:\n * Formula: \log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)\n * Proof: Using x = a^m$and$y = a^n:\n * Divide: \frac{x}{y} = \frac{a^m}{a^n} = a^{m-n}.\n * Convert to log form: \log_a\left(\frac{x}{y}\right) = m - n = \log_a(x) - \log_a(y).\n * **Logarithm Law for Powers (Power Rule)**:\n * Formula: \log_a(x^n) = n \log_a(x)\n\n* **Special Logarithmic Identities**\n * \log_a(1) = 0$(since$a^0 = 1)\n * \log_a(a) = 1$(since$a^1 = a)\n * \log_a\left(\frac{1}{x}\right) = \log_a(x^{-1}) = -\log_a(x)\n\n* **Comprehensive Worked Examples**\n * **Worked Example 26: Simplifying to Single Logarithm Terms**\n * Simplify \log_a(4) + \log_a(5)$:$\log_a(4 \times 5) = \log_a(20).\n * Simplify \log_a(22) - \log_a(11)$:$\log_a\left(\frac{22}{11}\right) = \log_a(2).\n * Simplify 3 \log_a(2)$:$\log_a(2^3) = \log_a(8).\n * **Worked Example 27: Simplifying and Evaluating Expressions**\n * Evaluate \log_2(1)$:$0.\n * Evaluate \log_5(5)$:$1.\n * Evaluate \log_6\left(\frac{1}{36}\right)$:$\log_6(6^{-2}) = -2 \log_6(6) = -2(1) = -2\n * Evaluate \log_2(6) - \log_2(3)$:$\log_2\left(\frac{6}{3}\right) = \log_2(2) = 1$.

Solving Exponential Equations Using Logarithms

• Algebraic Solution Techniques

  • Method 1: Direct Base Conversion

    • For equation $a^x = y$

    • Convert directly to logarithm: $x = \log_a(y)$

  • Method 2: Common Base 10 Conversion

    • For equation $a^x = y$

    • Take $\log_{10}$ of both sides: $\log_{10}(a^x) = \log_{10}(y)$

    • Apply power law: $x \log_{10}(a) = \log_{10}(y)$

    • Rearrange: $x = \frac{\log_{10}(y)}{\log_{10}(a)}$

  • Change of Base Formula:

    • $\log_a(y) = \frac{\log_b(y)}{\log_b(a)}$ (commonly using $b = 10$)

• Comprehensive Worked Examples

  • Worked Example 28: Direct Base Method

    • Solve $2^x = 7$ to $3$ decimal places:

      • $x = \log_2(7) \approx 2.807$

    • Solve $50 \times 1.1^x = 100$ to $3$ decimal places:

      • Divide by $50$: $1.1^x = 2$

      • Convert to log: $x = \log_{1.1}(2) \approx 7.273$

  • Worked Example 29: Base 10 Method

    • Solve $3^x = 5$ using base $10$ to $3$ decimal places:

      • $\log_{10}(3^x) = \log_{10}(5)$

      • $x \log_{10}(3) = \log_{10}(5)$

      • $x = \frac{\log_{10}(5)}{\log_{10}(3)} \approx 1.465$

    • Solve $1000 \times 0.93^x = 100$ using base $10$ to $3$ decimal places:

      • Divide by $1000$: $0.93^x = 0.1$

      • $\log_{10}(0.93^x) = \log_{10}(0.1)$

      • $x \log_{10}(0.93) = -1$

      • $x = \frac{-1}{\log_{10}(0.93)} \approx 31.729$