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: 1. **Define** a logarithm as the inverse of an exponential function 2.

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](/courses/bsma1001/notes/.%2Fweek06%2Flogarithms-equations) - Previous topic: [Exponential Functions](/courses/bsma1001/notes/.%2Fweek05%2Ffunctions-exponential) - 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

What this note is really saying

Logarithmic Functions and Properties

🎯 Learning Objectives

📋 Prerequisites

📖 Core Content

8.1 Intuition: The Inverse of Exponentiation

8.2 Definition

8.3 Laws of Logarithms

8.3.1 The Three Core Laws

8.3.2 Additional Properties

8.3.3 Change of Base Formula

8.4 Graphing Logarithmic Functions

8.4.1 Graph of f(x)=log⁡axf(x) = \log_a xf(x)=loga​x, a>1a > 1a>1

8.4.2 Graph of f(x)=log⁡axf(x) = \log_a xf(x)=loga​x, 0<a<10 < a < 10<a<1

8.5 Natural Logarithm ln⁡x\ln xlnx

8.6 Worked Examples

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

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

8.5 Natural Logarithm $\ln x$