Algebra II 4th Quarter Exam Review Notes

Fundamental Logarithmic Principles and Properties

• Simplification of Natural Logarithms

  • The term $\ln(e)$ involves the natural logarithm, which is a logarithm with base $e$ (Euler's number).

  • By definition, $\log_b(b) = 1$.

  • Therefore, $\ln(e) = \log_e(e) = 1$.

• Conversion Between Logarithmic and Exponential Form

  • The basic relationship is defined as $\log_b(x) = y \iff b^y = x$.

  • Case 1: Solving for the argument. If $\log_3(x) = 4$, we rewrite this in exponential form as $3^4 = x$. Calculating the power, we find $x = 81$.

  • Case 2: Solving for the argument with fractional exponents. If $\log_{81}(x) = \frac{1}{2}$, the exponential form is $81^{1/2} = x$. Since raising a number to the $\frac{1}{2}$ power is equivalent to taking the square root, $x = \sqrt{81} = 9$.

• Inverse Properties of Logarithms

  • The property $b^{\log_b(x)} = x$ states that an exponential base raised to a logarithm with the same base simplifies to the argument of the logarithm.

  • Applying this to the expression $2^{\log_2(7)} = x$, we see the bases match, resulting in $x = 7$.

• Euler's Number

  • The irrational number $e$ is a fundamental mathematical constant used in calculus and finance.

  • Its approximate numerical value is $e \approx 2.718$.

• Logarithmic Laws: The Quotient Property

  • The property states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: $\log\left(\frac{A}{B}\right) = \log(A) - \log(B)$.

  • For the expression $\log\left(\frac{a}{B}\right)$, it simplifies to $\log(a) - \log(B)$.

• Evaluating Logarithms with Change of Base Logic

  • To find $x$ in $\log_{27}(9) = x$, convert to exponential form: $27^x = 9$.

  • Express both sides with a common base of 3:

    • $(3^3)^x = 3^2$

    • $3^{3x} = 3^2$

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

Advanced Logarithmic and Exponential Problem Solving

• Estimating Logarithms Between Integers

  • To determine between which two integers $\log(50)$ lies, consider the common log (base 10).

  • Since $\log(10) = 1$ and $\log(100) = 2$, and the value 50 falls between 10 and 100 (10 < 50 < 100), it follows that $\log(50)$ is between the integers 1 and 2.

• Solving Exponential Equations Using Logarithms

  • To solve $4^x = 19$ for $x$, apply the logarithm to both sides: $\log(4^x) = \log(19)$.

  • Using the power property: $x \cdot \log(4) = \log(19)$.

  • Isolating $x$ gives $x = \frac{\log(19)}{\log(4)}$.

• Rational and Negative Exponents

  • Property: $a^{-n} = \frac{1}{a^n}$ and $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

  • Simplifying $36^{-1/2}$:

    • $36^{-1/2} = \frac{1}{36^{1/2}} = \frac{1}{\sqrt{36}} = \frac{1}{6}$.

  • Simplifying $81^{3/4}$:

    • $81^{3/4} = (\sqrt[4]{81})^3 = 3^3 = 27$.

  • Simplifying $7^{3.7}$:

    • This represents an irrational power of 7, solvable via calculator as $\approx 1345.54$.

• Solving Equations with Logarithms and Mixed Terms

  • Logarithmic Quadratic: $\log_3(x^2 - 7) = 4$.

    • Convert to exponential form: $x^2 - 7 = 3^4$.

    • $x^2 - 7 = 81 \implies x^2 = 88$.

    • $x = \pm \sqrt{88} = \pm 2\sqrt{22}$.

  • Exponential with Function Bases: $5^x = 25^{x+2}$.

    • Convert to a common base: $5^x = (5^2)^{x+2}$.

    • $5^x = 5^{2x+4}$.

    • Equate exponents: $x = 2x + 4 \implies -x = 4 \implies x = -4$.

Arithmetic and Geometric Sequences

• Arithmetic Sequences

  • Identification: A sequence is arithmetic if it has a constant common difference ($d$) between terms.

  • For the sequence $12, 7, 2, -3$, calculating the difference: $7 - 12 = -5$, $2 - 7 = -5$. Since the difference is constant, it is an arithmetic sequence.

  • Next Term Calculation: To find the next term in $11, 6, 1, -4$, subtract the common difference ($d = -5$) from the last term: $-4 - 5 = -9$.

  • Finding the $n$-th Term: The formula is $a_n = a_1 + (n-1)d$.

    • For a sequence with $a_1 = 4$ and $d = 5$, the 30th term is: $a_{30} = 4 + (30-1)5 = 4 + (29)(5) = 4 + 145 = 149$.

  • Finding the Common Difference from Points: Given $a_1 = 7$ and $a_8 = 35$.

    • Use $a_n = a_1 + (n-1)d \implies 35 = 7 + (8-1)d$.

    • $28 = 7d \implies d = 4$.

• Geometric Sequences

  • Ratio Identification: The common ratio ($r$) is found by dividing a term by its predecessor: $r = \frac{a_n}{a_{n-1}}$.

  • Finding $r$ from non-consecutive terms: Given $a_1 = -54$ and $a_4 = 2$.

    • Formula: $a_4 = a_1 \cdot r^{4-1} \implies 2 = -54 \cdot r^3$.

    • Divide by $-54$: $r^3 = -\frac{2}{54} = -\frac{1}{27}$.

    • Take the cube root: $r = \sqrt[3]{-\frac{1}{27}} = -\frac{1}{3}$.

  • Finding the $n$-th Geo metric Term: The formula is $a_n = a_1 \cdot r^{n-1}$.

    • In the sequence $3, 1, \frac{1}{3}, ...$, the first term $a_1 = 3$ and ratio $r = \frac{1}{3}$.

    • The 7th term is: $a_7 = 3 \cdot \left(\frac{1}{3}\right)^{7-1} = 3 \cdot \left(\frac{1}{3}\right)^6 = 3 \cdot \frac{1}{729} = \frac{1}{243}$.

Financial Formulas, Roots, and Variations

• Compound Interest

  • The formula for discrete compounding is $A = P\left(1 + \frac{r}{n}\right)^{nt}$.

  • Parameters:

    • $P$ (Principal) = $600$

    • $r$ (Annual Interest Rate) = $0.08$ (8%)

    • $n$ (Compounding periods per year) = $4$ (Quarterly)

    • $t$ (Time in years) = $5$

  • Formula Setup: $V = 600\left(1 + \frac{0.08}{4}\right)^{4 \times 5}$.

  • Simplified Formula: $V = 600(1.02)^{20}$.

• Simplifying Radicals with Variables

  • To simplify $\sqrt[n]{x^m}$, use the power rule $(x^m)^{1/n} = x^{m/n}$.

  • Expression: $\sqrt[3]{x^6}$.

  • Calculation: $x^{6/3} = x^2$.

• Direct Proportionality (Variation)

  • The formula is $s = kt$, where $k$ is the constant of variation.

  • Step 1: Find $k$. If $s = 35$ when $t = 5$, then $35 = k(5) \implies k = 7$.

  • Step 2: Solve for target variable. To find $t$ when $s = 91$, use $91 = 7t \implies t = 13$.

• Joint Proportionality (Variation)

  • The formula is $z = kxy$, where $z$ varies jointly with $x$ and $y$.

  • Step 1: Find $k$. If $z = 24$ when $x = 3$ and $y = 4$:

    • $24 = k(3)(4) \implies 24 = 12k \implies k = 2$.

  • Step 2: Solve for target variable. To find $z$ when $x = 5$ and $y = 6$:

    • $z = 2(5)(6) = 60$.