Comprehensive Study Guide on Surds and Logarithms

Definition and Fundamental Rules of Surds

• A surd is an irrational number of the form $\sqrt[n]{x}$, where $n$ is a positive integer that is not a perfect square.

• Examples of surds and non-surds include:

  • $\sqrt{1} = 1$ is not a surd because $1$ is a perfect square.

  • $\sqrt{2} = 1.214...$ is a surd because $2$ is not a perfect square.

  • $2\sqrt{3} = 1.732...$ is a surd.

  • $\sqrt{4} = 2$ is not a surd because $4$ is a perfect square.

• The fundamental rules of surds allow for the simplification and manipulation of these expressions:

  1. Multiplication Rule: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

  2. Division Rule: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

  3. Roots of a Square: $\sqrt{a} \times \sqrt{a} = a$

  4. Addition and Subtraction Rule: These operations apply only to "like surds."

     • $\sqrt{a} + \sqrt{a} = 2\sqrt{a}$

     • $3\sqrt{a} - 2\sqrt{a} = \sqrt{a}$

• Bell work concepts associated with surds include perfect squares and square roots.

Simplifying Surds and Arithmetic Operations

• Simplifying surds involves breaking down a radicand into its factors to find perfect squares that can be extracted from the root.

• Work Example (1): Simplify the following:

  • $\sqrt{12}$: $\sqrt{4 \times 3} = 2\sqrt{3}$

  • $\frac{\sqrt{20}}{2}$: $\frac{\sqrt{4 \times 5}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}$

  • $5\sqrt{6} - 2\sqrt{24} + \sqrt{54}$: Simplifying each term results in $5\sqrt{6} - 2\sqrt{4 \times 6} + \sqrt{9 \times 6} = 5\sqrt{6} - 4\sqrt{6} + 3\sqrt{6} = 4\sqrt{6}$

• Multiplication, Division, and Expansion Examples:

  • $\sqrt{3} \times \sqrt{5} = \sqrt{15}$

  • $(\sqrt{8})^2 = 8$

  • $2\sqrt{5} \times 3\sqrt{3} = 6\sqrt{15}$

• Expansion and Simplification:

  • $(4 - \sqrt{3})^2 = (4 - \sqrt{3})(4 - \sqrt{3}) = 16 - 4\sqrt{3} - 4\sqrt{3} + 3 = 19 - 8\sqrt{3}$

  • $(\sqrt{3} + 5\sqrt{2})(\sqrt{2} + \sqrt{3}) = \sqrt{6} + 3 + 10 + 5\sqrt{6} = 13 + 6\sqrt{6}$

• Further Simplifying Examples:

  • $\sqrt{75} - \sqrt{12} = \sqrt{25 \times 3} - \sqrt{4 \times 3} = 5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}$

  • $\sqrt{11}$ remains in its surd form.

Rationalizing the Denominator and Conjugate Surds

• Rationalizing a denominator involves removing any surds from the bottom of a fraction so that the denominator is a rational number.

• Conjugate surds are pairs like $4 + 2\sqrt{3}$ and $4 - 2\sqrt{3}$. Multiplying a surd expression by its conjugate results in a rational number.

• Work Example (4): Rationalizing the denominators of:

  • $\frac{2}{\sqrt{5}}$: Multiply numerator and denominator by $\sqrt{5}$ to get $\frac{2\sqrt{5}}{5}$.

  • $\frac{5}{2 + \sqrt{5}}$: Multiply by the conjugate $(2 - \sqrt{5})$ to get $\frac{5(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} = \frac{10 - 5\sqrt{5}}{4 - 5} = \frac{10 - 5\sqrt{5}}{-1} = 5\sqrt{5} - 10$.

  • $\frac{7 + 3\sqrt{5}}{\sqrt{5} - \sqrt{2}}$: Multiply by the conjugate $(\sqrt{5} + \sqrt{2})$ to get $\frac{(7 + 3\sqrt{5})(\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} = \frac{7\sqrt{5} + 7\sqrt{2} + 15 + 3\sqrt{10}}{5 - 2} = \frac{7\sqrt{5} + 7\sqrt{2} + 15 + 3\sqrt{10}}{3}$.

Geometric and Word Problems Involving Surds

• Area and Dimensions:

  • A garden is $\sqrt{30}\,\text{m}$ long and $8\,\text{m}$ wide. A pond within it is $\sqrt{2}\,\text{m}$ long and $\sqrt{6}\,\text{m}$ wide. The area of the pond is expressed as a percentage of the total garden area.

  • Rectangle sides are $(2 + \sqrt{8})\,\text{cm}$ and $(7 - \sqrt{2})\,\text{cm}$. The area in the form $a + b\sqrt{2}$ is calculated by expanding the product of the side lengths.

• Trigonometry and Geometry:

  • In a triangle with vertices $A, B, X$:

    • $AC^2$ is calculated using the Pythagorean theorem with given surd lengths.

    • $\tan(x)$ and the triangle area are expressed in the form $\frac{a\sqrt{b}}{b}$ and $\frac{p\sqrt{6}}{q}$.

• Volume of a Cuboid:

  • A cuboid with a square base of side length $(1 + \sqrt{2})\,\text{cm}$ and height $(5 - \sqrt{2})\,\text{cm}$ has a volume found by $\text{Area of base} \times \text{height} = (1 + \sqrt{2})^2 \times (5 - \sqrt{2})$. The final answer is in the form $a + b\sqrt{2}$.

• Height of a Cylinder:

  • A right circular cylinder has a volume of $(25 + 14\sqrt{3})\pi\,\text{cm}^3$ and a base radius of $(2 + \sqrt{3})\,\text{cm}$. The height $h$ is found using $V = \pi r^2 h$, solved in the form $(a + b\sqrt{3})\,\text{cm}$.

Introduction to Logarithms to Base 10

• If an exponential relationship is given as $y = 10^x$, then the logarithmic form is $x = \log_{10}(y)$.

• Fundamental properties of base 10 logarithms:

  • $\log_{10}(10) = 1$

  • $\log_{10}(1) = 0$

• Conversions and Solving:

  • Exponential to Logarithmic: $10^{1.653} = 45 \rightarrow \log_{10}(45) = 1.653$

  • Logarithmic to Exponential: $\log_{10}(x) = 2.9 \rightarrow x = 10^{2.9}$. Correct to $3$ significant figures, $x \approx 794$.

• Evaluating Base 10 Logarithms:

  • $\log_{10}(100000) = 5$

  • $\log_{10}(0.001) = -3$

  • $\log_{10}(100\sqrt{10}) = \log_{10}(10^{2} \times 10^{0.5}) = \log_{10}(10^{2.5}) = 2.5$

Logarithms to Base a and General Laws

• The general definition of a logarithm to base $a$ is: if $y = a^x$, then $x = \log_a(y)$.

• Basic Logarithmic Identities:

  • $\log_a(a) = 1$

  • $\log_a(1) = 0$

  • $\log_a(a^x) = x$

  • $x = a^{\log_a(x)}$

• Laws of Logarithms:

  1. Multiplication Law: $\log_a(xy) = \log_a(x) + \log_a(y)$

  2. Division Law: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

  3. Power Law: $\log_a(x^k) = k\log_a(x)$

  4. Negative Power Property: $\log_a\left(\frac{1}{x}\right) = -\log_a(x)$

• Change of Base Rule:

  • $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

  • $\log_b(a) = \frac{1}{\log_a(b)}$

Solving Logarithmic and Exponential Equations

• Techniques for solving logarithmic equations involve using the laws to combine terms into a single logarithm or converting to exponential form.

• Work Example (12):

  • Solve $2\log_5(x + 2) = \log_5(2x + 19)$: Apply power law to get $\log_5((x + 2)^2) = \log_5(2x + 19)$. This implies $(x + 2)^2 = 2x + 19$.

  • Solve $4\log_x(2) - \log_x(4) = 8$.

• Solving Exponential Equations:

  • Use logarithms when bases are different: For $3^x = 40$, take logs of both sides: $x\log(3) = \log(40)$, so $x = \frac{\log(40)}{\log(3)} \approx 3.36$.

  • Quadratic substitution: Equations like $5^{2x} + 7(5^x) - 30 = 0$ can be solved by letting $u = 5^x$, resulting in $u^2 + 7u - 30 = 0$.

• Equations with Natural Logarithms:

  • The natural logarithm has base $e \approx 2.718$. If $y = e^x$, then $x = \ln(y)$.

  • $\ln(e^x) = x$

  • $e^{\ln(x)} = x$

Graphs of Exponents and Logarithms

• Exponential Graphs ($y = a^x$):

  • These graphs pass through $(0, 1)$. For $a > 1$, the function is increasing. As $x \rightarrow -\infty$, $y \rightarrow 0$ (horizontal asymptote).

  • Graphs like $y = 3^x, y = 2^x, y = 1.5^x$ all intersect at $(0, 1)$ but have different steepness.

• Logarithmic Graphs ($y = \log_a(x)$):

  • The graph of $y = \log_a(x)$ is the reflection of $y = a^x$ in the line $y = x$.

  • Logarithmic graphs pass through $(1, 0)$ and have a vertical asymptote at $x = 0$.

• Natural Logarithm and e Graphs:

  • $y = e^x$ and $y = e^{-x}$ are reflections of each other in the y-axis.

  • $y = e^x$ is the inverse of $y = \ln(x)$.

• Transformations:

  • $y = e^x + 1$ is a vertical shift up by $1$.

  • $y = 10 e^{-x}$ involves a vertical stretch and reflection across the y-axis.

  • $y = \ln(x + 1)$ is a horizontal shift left by $1$, moving the vertical asymptote to $x = -1$.

Practical Exercises and Numerical Methods

• Solving equations graphically involves finding the intersection of a curve and a line.

• Example: Solve $e^x = 8$ by drawing $y = e^x - 2$ and finding where it intersects $y = 6$.

• Numerical Tables: Generating values for expressions such as $y = 5\sin(2x) - 2\cos(x)$ to plot graphs and find intersections with lines like $y = 0$ or other trigonometric functions.

• Use of Calculators: Natural logarithms and powers of $e$ often require evaluation to $3$ significant figures for accuracy in practical exercises.