Grade 12 Mathematics
Grade 12 · Term 1Mathematics

Functions & Inverses

We study the logarithmic function as the inverse of the exponential, and apply logarithm laws to simplify, solve equations, and sketch graphs.

Week 3

2.1 Logarithms & the Log Function

  • Define $\log_a x$ as the inverse of $a^x$
  • Apply log laws: product, quotient, power
  • Sketch $y=\log_a x$ and understand transformations
  • Solve exponential and log equations
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Real-World Connection

The Richter scale for earthquakes uses logarithms: a magnitude 7 quake is 10 times stronger than a magnitude 6. Sound decibels, pH acidity, and even stellar brightness all use logarithmic scales — they compress huge ranges into manageable numbers.

Definition

Logarithm

The logarithm log⁡ax=y\log_a x = y means ay=xa^y = x. It answers the question: 'To what power must I raise aa to get xx?'

log⁡ax=y⇔ay=x(a>0,a≠1,x>0)\log_a x = y \Leftrightarrow a^y = x \quad (a>0, a\neq1, x>0)

Product Rule

log⁡a(xy)=log⁡ax+log⁡ay\log_a(xy) = \log_a x + \log_a y

$x,y>0$

Quotient Rule

log⁡a(xy)=log⁡ax−log⁡ay\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y

$x,y>0$

Power Rule

log⁡a(xn)=nlog⁡ax\log_a(x^n) = n\log_a x

$x>0$

Change of Base

log⁡ax=log⁡b⋅xlog⁡ba=ln⁡xln⁡a\log_a x = \frac{\log b \cdot x}{\log_b a} = \frac{\ln x}{\ln a}

Convert to any convenient base

Property / Rule

Graph of $y=\log_a x$

Domain: x>0x>0. Range: all reals. xx-intercept at (1,0)(1,0). If a>1a>1: increasing. If 0<a<10<a<1: decreasing. Vertical asymptote: x=0x=0 (the yy-axis). No yy-intercept.

Asymptote: x=0;f−1(x)=ax\text{Asymptote: }x=0;\quad f^{-1}(x)=a^x

🚨 Common Mistake

Common error: log⁡(x+y)≠log⁡x+log⁡y\log(x+y)\neq\log x+\log y. The product rule applies to a LOG of a PRODUCT, not to a sum inside the log. log⁡(x+y)\log(x+y) cannot be simplified further.

Worked Examples

Worked Example

Solve an exponential equation using logs

Problem

Solve: 5^x = 80

Worked Example

Simplify using log laws

Problem

Simplify:log⁡296−log⁡26+log⁡214Simplify: \log_2 96 - \log_2 6 + \log_2 \frac{1}{4}
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge1 mark
Write log⁡381=4\log_3 81=4 in exponential form.
2
L1 · Knowledge2 marks
Evaluate log⁡100.001\log_{10}0.001.
3
L2 · Routine Procedures3 marks
Solve log⁡x64=3\log_x 64=3.
4
L2 · Routine Procedures2 marks
Simplify log⁡2+log⁡50\log 2+\log 50 (base 10).
5
L3 · Complex Procedures5 marks
Solve log⁡3(x+1)+log⁡3(x−1)=3\log_3(x+1)+\log_3(x-1)=3.
6
L3 · Complex Procedures4 marks
Sketch y=log⁡2xy=\log_2 x and y=log⁡1/2xy=\log_{1/2} x on the same axes, showing key points.
7
L4 · Problem Solving3 marks
Without a calculator, find: log⁡52+log⁡512.5\log_5 2 + \log_5 12.5.
8
L4 · Problem Solving5 marks
Solve 2x+1+2x=482^{x+1}+2^x=48.

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Functions & Inverses Grade 12 Maths CAPS Notes & Examples | MathSciBuddy