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Logarithmic Functions and Properties

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Logarithmic Functions and Properties

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Logarithmic Functions and Properties

🎯 Learning Objectives

By the end of this topic, you will be able to:

Define a logarithm as the inverse of an exponential function
Apply the laws of logarithms (product, quotient, power, change of base)
Graph logarithmic functions for $a > 1$ and $0 < a < 1$
Convert between exponential and logarithmic forms
Use the natural logarithm $\ln x$ and common logarithm $\log x$

📋 Prerequisites

Exponential Functions — properties of exponentials
Inverse Functions — concept of inverse functions

📖 Core Content

8.1 Intuition: The Inverse of Exponentiation

A logarithm answers the question: "What exponent gives me this number?" If

2^3 = 8

, then

\log_2 8 = 3

("log base 2 of 8 equals 3"). The logarithm undoes exponentiation. Think of it this way:

Exponential: $b^x = y$ asks: "If I multiply $b$ by itself $x$ times, what do I get?"
Logarithm: $\log_b y = x$ asks: "How many times do I multiply $b$ to get $y$ ?"

💡 Why this matters: Logarithms convert multiplicative relationships into additive ones. This is used everywhere: measuring earthquakes (Richter scale), sound (decibels), acidity (pH), algorithmic complexity ( $\log n$ is the holy grail of efficiency), and in ML (log-likelihood, logistic regression).

8.2 Definition

For

b > 0

b \neq 1

, and

x > 0

\log_b x = y \iff b^y = x

Base $b$ : the base of the exponential
Argument $x$ : the value whose logarithm we take ( $x > 0$ )
Result $y$ : the exponent Special bases:

Base	Name	Notation
10	Common logarithm	$\log_{10} x$ or simply $\log x$
$e$	Natural logarithm	$\log_e x$ or $\ln x$
2	Binary logarithm	$\log_2 x$ (important in CS)

Key relationship:

\log_b x \text{ and } b^x \text{ are inverse functions}

b^{\log_b x} = x \quad \text{and} \quad \log_b(b^x) = x

(Diagram)

8.3 Laws of Logarithms

8.3.1 The Three Core Laws

Law	Formula	Example	Why
Product	$\log_b(xy) = \log_b x + \log_b y$	$\log_2(8 \cdot 4) = 3 + 2 = 5$	$b^{m+n} = b^m \cdot b^n$
Quotient	$\log_b(x/y) = \log_b x - \log_b y$	$\log_3(81/9) = 4 - 2 = 2$	$b^{m-n} = b^m / b^n$
Power	$\log_b(x^n) = n \cdot \log_b x$	$\log(100^3) = 3 \cdot 2 = 6$	$(b^m)^n = b^{mn}$

Derivation of Product Law: Let

m = \log_b x

and

n = \log_b y

. Then

b^m = x

and

b^n = y

. Then

xy = b^m \cdot b^n = b^{m+n}

. Taking

\log_b

of both sides:

\log_b(xy) = m + n = \log_b x + \log_b y

. ∎

8.3.2 Additional Properties

Property	Formula	Example
Identity	$\log_b 1 = 0$	$\log_5 1 = 0$ (since $5^0 = 1$ )
Base	$\log_b b = 1$	$\log_7 7 = 1$ (since $7^1 = 7$ )
Reciprocal	$\log_b(1/x) = -\log_b x$	$\log_2(1/8) = -3$
Change of base	$\log_b a = \frac{\log_c a}{\log_c b}$	$\log_2 5 = \frac{\log 5}{\log 2}$

8.3.3 Change of Base Formula

\log_b a = \frac{\log_c a}{\log_c b}

Why this matters: Your calculator probably only has

\log

(base 10) and

\ln

(base

e

). To compute

\log_2 5

, you use:

\log_2 5 = \frac{\ln 5}{\ln 2} \approx \frac{1.609}{0.693} \approx 2.322

8.4 Graphing Logarithmic Functions

8.4.1 Graph of $f(x) = \log_a x$ , $a > 1$

For

f(x) = \log_2 x

$x$	$1/8$	$1/4$	$1/2$	$1$	$2$	$4$	$8$
$f(x)$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$

Properties:

Domain: $(0, \infty)$ (only positive inputs)
Range: $\mathbb{R}$ (all real outputs)
$x$ -intercept: $(1, 0)$
Vertical asymptote: $x = 0$ (y-axis)
Increasing (for $a > 1$ )

8.4.2 Graph of $f(x) = \log_a x$ , $0 < a < 1$

For

f(x) = \log_{1/2} x

Decreasing (since base $< 1$ )
Same intercept at $(1, 0)$
Same asymptote $x = 0$

🔍 Note: $\log_{1/a} x = -\log_a x$ . So decay-base logs are reflections of growth-base logs across the $x$ -axis. (Diagram)

8.5 Natural Logarithm $\ln x$

The natural logarithm

\ln x = \log_e x

has base

e \approx 2.718

. It's the inverse of

e^x

\ln(e^x) = x \quad \text{and} \quad e^{\ln x} = x

Properties (same as any log, but with base

e

$\ln 1 = 0$
$\ln e = 1$
$\ln(xy) = \ln x + \ln y$
$\ln(x/y) = \ln x - \ln y$
$\ln(x^n) = n \ln x$

8.6 Worked Examples

Example 1.1 (Easy): Evaluate

\log_3 81

. Step 1:

81 = 3^4

Step 2:

\log_3 81 = \log_3(3^4) = 4

\boxed{4}

Example 1.2 (Medium): Expand

\log_2\left(\frac{8x^3}{y}\right)

. \log_2\left(\frac{8x^3}{y}\right) &= \log_2(8x^3) - \log_2 y \\ &= \log_2 8 + \log_2(x^3) - \log_2 y \\ &= 3 + 3\log_2 x - \log_2 y \end{aligned}

\\boxed{3 + 3\\log_2 x - \\log_2 y} **Example 1.3 (Hard):** Condense $2\\ln x - \\frac{1}{2}\\ln(x^2+1) + \\ln 3$ into a single logarithm.

\begin{aligned} &= \ln(x^2) - \ln\sqrt{x^2+1} + \ln 3 \\ &= \ln\left(\frac{x^2}{\sqrt{x^2+1}}\right) + \ln 3 \\ &= \ln\left(\frac{3x^2}{\sqrt{x^2+1}}\right) \end{aligned}

\\boxed{\\ln\\left(\\frac{3x^2}{\\sqrt{x^2+1}}\\right)} **Example 1.4 (Change of base):** Compute $\\log_4 32$ to 3 decimal places. \\log_4 32 = \\frac{\\log_{10} 32}{\\log_{10} 4} = \\frac{1.50515}{0.60206} \\approx 2.500 (Check: $4^{2.5} = 4^{5/2} = \\sqrt{4^5} = \\sqrt{1024} = 32$ ✓) \\boxed{2.500} \--- ## 📐 Key Formulas — Summary Table | Concept | Formula | Notes | |:---|:---|:---| | Definition | $\\log_b x = y \\iff b^y = x$ | $b > 0, b \\neq 1, x > 0$ | | Product | $\\log_b(xy) = \\log_b x + \\log_b y$ | | | Quotient | $\\log_b(x/y) = \\log_b x - \\log_b y$ | | | Power | $\\log_b(x^n) = n \\log_b x$ | | | Change of base | $\\log_b a = \\frac{\\log_c a}{\\log_c b}$ | For calculator use | | Identity | $\\log_b 1 = 0$ | | | Base | $\\log_b b = 1$ | | | Natural log | $\\ln x = \\log_e x$ | | | Inverse | $b^{\\log_b x} = x$, $\\log_b(b^x) = x$ | | | Domain | $(0, \\infty)$ | | | Range | $\\mathbb{R}$ | | | $x$-intercept | $(1, 0)$ | | | Asymptote | $x = 0$ (vertical) | | --- ## ⚠️ Common Pitfalls ### Pitfall 1: $\\log_b(x+y) \\neq \\log_b x + \\log_b y$ **Mistake:** Thinking the product law applies to addition. **Correct:** Only **multiplication** becomes addition: $\\log_b(xy) = \\log_b x + \\log_b y$. There's no simple rule for $\\log_b(x+y)$. ### Pitfall 2: $\\log_b(x^n) \\neq (\\log_b x)^n$ **Mistake:** Confusing power inside argument with power outside. **Correct:** $\\log_b(x^n) = n \\log_b x$ (bring exponent in front). Not $(\\log_b x)^n$. ### Pitfall 3: Forgetting the Domain **Mistake:** Taking $\\log x$ for $x \\leq 0$. **Correct:** $\\log_b x$ is only defined for $x > 0$. $\\log(-5)$ is undefined (in real numbers). ### Pitfall 4: Misapplying Change of Base **Mistake:** Writing $\\log_2 5 = \\frac{\\log 2}{\\log 5}$ (swapped numerator/denominator). **Correct:** $\\log_2 5 = \\frac{\\log 5}{\\log 2}$. The original argument goes in the numerator. --- ## 📝 Practice Questions > **Q1: Evaluate $\\log_5 125$.** > > **Strategy Hint:** $125 = 5^3$. > >

\boxed{3}

> **Q2: Evaluate $\\ln(e^7)$.** > > $\\ln(e^7) = 7$ > >

\boxed{7}

> **Q3: Expand $\\log\\left(\\frac{100x^2}{y^3}\\right)$.** > > $= \\log 100 + 2\\log x - 3\\log y = 2 + 2\\log x - 3\\log y$ > >

\boxed{2 + 2\log x - 3\log y}

> **Q4: Condense $3\\ln x + \\ln y - 2\\ln z$.** > > $= \\ln(x^3) + \\ln y - \\ln(z^2) = \\ln\\left(\\frac{x^3 y}{z^2}\\right)$ > >

\boxed{\ln\left(\frac{x^3 y}{z^2}\right)}

> **Q5: Compute $\\log_3 27$ without a calculator.** > > $27 = 3^3$, so $\\log_3 27 = 3$ > >

\boxed{3}

> **Q6: Compute $\\log_5 2$ using change of base (to 3 decimal places).** > > $\\log_5 2 = \\frac{\\ln 2}{\\ln 5} \\approx \\frac{0.693}{1.609} \\approx 0.431$ > >

\boxed{0.431}

> **Q7: Solve $\\log_2(x) = 5$.** > > $x = 2^5 = 32$ > >

\boxed{32}

> **Q8: What is the domain of $f(x) = \\log_3(x-2)$?** > > Need $x-2 > 0 \\to x > 2$ > >

\boxed{(2, \infty)}

> **Q9: Simplify $\\log_2 8 + \\log_2 16$.** > > $\\log_2 8 = 3$, $\\log_2 16 = 4$, sum = 7 > >

\boxed{7}

> **Q10: Write in exponential form: $\\log_3 81 = 4$.** > > $3^4 = 81$ > >

\boxed{3^4 = 81}

> **Q11: Find $x$: $\\log_x 125 = 3$.** > > $x^3 = 125 \\to x = \\sqrt\[3\]{125} = 5$ > >

\boxed{5}

> **Q12: Show that $\\log_a b = \\frac{1}{\\log_b a}$.** > > **Strategy Hint:** Use change of base with $c = b$. > > $\\log_a b = \\frac{\\log_b b}{\\log_b a} = \\frac{1}{\\log_b a}$ > >

\boxed{\text{Proved}}$$ --- ## 🔗 Cross-References - Next topic: [Exponential & Logarithmic Equations](../week06/logarithms-equations.md) - Previous topic: [Exponential Functions](../week05/functions-exponential.md) - Across courses: BSMA1002 Stats 1 (log transformations in regression); BSMA1003 Maths 2 (logarithmic differentiation) Join Discord Previous5.2 Composite & Inverse FunctionsNext6.2 Exponential & Log Equations