Logarithms

Subject: Additional Mathematics
Topic: 5
Cambridge Code: 4037 / 0606

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Definition of Logarithm

Logarithm - The inverse function of an exponential function

Relationship to Exponentials

If $a^x = b$, then $\log_a b = x$

In other words: $\log_a b = x \iff a^x = b$

Examples

• $\log_2 8 = 3$ because $2^3 = 8$ ✓
• $\log_{10} 100 = 2$ because $10^2 = 100$ ✓
• $\log_5 25 = 2$ because $5^2 = 25$ ✓

Key Points

• $a$ is the base (must be positive, $a \neq 1$)
• $b$ is the argument (must be positive)
• $x$ is the logarithm (can be any real number)

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Common Logarithms

Base 10 Logarithm

$\log b = \log_{10} b$

Usually written without the base (implied base 10)

Natural Logarithm

$\ln b = \log_e b$

where $e \approx 2.718$ (base of natural logarithms)

Examples

• $\log 100 = \log_{10} 100 = 2$
• $\log 1 = 0$ (any base)
• $\ln e = 1$

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Laws of Logarithms

Law 1: Product Rule

$\log_a(xy) = \log_a x + \log_a y$

Example: $\log_2(8 \times 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5$ ✓ (Check: $8 \times 4 = 32 = 2^5$)

Law 2: Quotient Rule

$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$

Example: $\log_{10}\left(\frac{100}{10}\right) = \log_{10} 100 - \log_{10} 10 = 2 - 1 = 1$ ✓ (Check: $\frac{100}{10} = 10$)

Law 3: Power Rule

$\log_a(x^n) = n \log_a x$

Example: $\log_2(8^2) = 2\log_2 8 = 2 \times 3 = 6$ ✓ (Check: $8^2 = 64 = 2^6$)

Law 4: Change of Base

$\log_a x = \frac{\log_b x}{\log_b a}$

Or more commonly: $\log_a x = \frac{\ln x}{\ln a} \text{ or } \log_a x = \frac{\log x}{\log a}$

Example: Find $\log_5 8$ $\log_5 8 = \frac{\ln 8}{\ln 5} = \frac{2.079}{1.609} \approx 1.292$

Special Cases

$\log_a 1 = 0 \quad \text{(any base)}$ $\log_a a = 1 \quad \text{(any base)}$ $a^{\log_a x} = x$ $\log_a(a^x) = x$

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Solving Logarithmic Equations

Type 1: Direct Application

Equations where you apply the definition

Example: Solve $\log_2 x = 5$

$x = 2^5 = 32$

Type 2: Using Laws of Logarithms

Example: Solve $\log x + \log 4 = 2$

$\log(4x) = 2$ $4x = 10^2 = 100$ $x = 25$

Type 3: Complex Equations

Example: Solve $2\ln x - \ln 4 = \ln 8$

$\ln(x^2) - \ln 4 = \ln 8$ $\ln\left(\frac{x^2}{4}\right) = \ln 8$ $\frac{x^2}{4} = 8$ $x^2 = 32$ $x = \sqrt{32} = 4\sqrt{2}$ (taking positive root)

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Solving Exponential Equations

Use logarithms to solve equations with variables in exponent

Example 1: Solve $3^x = 20$

Taking logarithm of both sides: $\log(3^x) = \log 20$ $x \log 3 = \log 20$ $x = \frac{\log 20}{\log 3} = \frac{1.301}{0.477} \approx 2.727$

Example 2: Solve $2^{x+1} = 7$

$\log(2^{x+1}) = \log 7$ $(x+1)\log 2 = \log 7$ $x+1 = \frac{\log 7}{\log 2} = \frac{0.845}{0.301} \approx 2.807$ $x \approx 1.807$

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Logarithmic Functions and Graphs

Properties of $f(x) = \log_a x$ (where $a > 1$)

• Domain: $x > 0$
• Range: All real numbers
• Asymptote: Vertical line at $x = 0$
• x-intercept: $(1, 0)$
• Increasing: As $x$ increases, $y$ increases
• Inverse: $f^{-1}(x) = a^x$ (exponential)

Graph Features

• Passes through $(1, 0)$: $\log_a 1 = 0$
• Passes through $(a, 1)$: $\log_a a = 1$
• Curve above x-axis for $x > 1$
• Curve below x-axis for $0 < x < 1$
• Vertical asymptote at $x = 0$

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Key Points to Remember

1. Logarithm is inverse of exponential
2. $\log_a b = x$ means $a^x = b$
3. Four main laws: product, quotient, power, change of base
4. Logarithms only defined for positive arguments
5. Change of base to evaluate logs with any base
6. Use logs to solve exponential equations

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Worked Examples

Example 1: Simplify Using Laws

Simplify $\log_3 27 + \log_3 9 - \log_3 3$

$= \log_3(27 \times 9 \div 3) = \log_3(81) = \log_3(3^4) = 4$

Example 2: Solve Logarithmic Equation

Solve $\log(2x-1) = \log 5 + \log 3$

$\log(2x-1) = \log 15$ $2x - 1 = 15$ $x = 8$

Example 3: Solve Exponential Equation

Solve $5^x = 100$ (give answer to 3 d.p.)

$\ln(5^x) = \ln 100$ $x \ln 5 = \ln 100$ $x = \frac{\ln 100}{\ln 5} = \frac{4.605}{1.609} = 2.861$

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Practice Questions

1. Evaluate:

   • $\log_2 32$
   • $\log_4 2$
   • $\log_{27} 3$

2. Simplify:

   • $\log 50 + \log 2 - \log 5$
   • $2\ln 3 + \ln 2 - \ln 6$

3. Solve:

   • $\log_x 16 = 4$
   • $2^x = 50$
   • $\ln(x+1) + \ln 2 = \ln 8$

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Revision Tips

• Learn the four laws thoroughly
• Remember domain restriction: argument must be positive
• Use change of base for any base
• Logarithms useful for solving exponential equations
• Graphs are inverse of exponential functions