Exponential and Logarithmic Functions

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

Recall: An exponential function 𝑦 = 𝑎. 𝑏𝑥 is defined for 𝑏 > 0; 𝑏 ≠ 1.
(A revision of Exponential functions can be found on pages 55-58 of your text book)

Considering only exponential functions with 𝑎 = 1 for the moment, the inverse function of
𝑦 = b𝑥 is 𝑥 = by, but we do not yet have the mathematics to write 𝑥 = by in the form
𝑦 =…

The logarithmic function is a new function used to describe the inverse of the exponential
function i.e. we can use logarithms (called logs for short) to make the exponent the subject
of the equation.

𝑦 = log𝑏𝑥 means exactly the same as 𝑥 = 𝑏𝑦

e.g.1 Consider the function 𝑦 = 2𝑥:

function: 𝑦 = 2𝑥:
Inverse function (with 𝑥 the subject): 𝑥 = 2𝑦
Logarithmic form (with 𝑦 the subject): 𝑦 = log2 𝑥

In function notation:
Function 𝑓(𝑥) = 2𝑥
Inverse function 𝑓−1(𝑥) = log2𝑥

As graphs:




y = log2 x

Note:
 The graphs reflect in
the line 𝑦 = 𝑥.
y = 2x  The 𝑦-intercept of
𝑦 = 2𝑥 becomes the
𝑥-intercept of
𝑦 = log2𝑥.
 The asymptote of
𝑦 = 2𝑥 is the 𝑥-axis.
The asymptote of
𝑦 = log2𝑥 is the
𝑦-axis
y=x




1

, e.g.2 Function: f ( x )=¿𝑦 i.e. y=¿
Inverse function (with 𝑥 the subject): 𝑥 = ( 1)
2
Logarithmic form (with 𝑦 the subject): 𝑦 = 𝑙𝑜𝑔1 𝑥
2
In function notation: 𝑓−1(𝑥) = 𝑙𝑜𝑔1 𝑥
2


As graphs:


y=¿

y=x Note:
The graphs reflect in
the line 𝑦 = 𝑥.
The 𝑦-intercept in
1 𝑥
𝑦 = ( ) becomes the
2
𝑥-intercept of
𝑦 = log1𝑥.
2
 The asymptote
1 𝑥
of
𝑦
2
= ( ) is the 𝑥-axis.
The asymptote
of
𝑦 = log1𝑥 Is the
𝑦 = log1 𝑥 2
2 𝑦-axis




Let’s take a closer look at LOGARITHMS

We need to be able to convert from exponential form to log form and vice versa.




LOG FORM log𝑎𝑏 = 𝑥

number bas Logarithm/exponent
e

EXPONENTIAL FORM 𝑎𝑥 = 𝑏


2