Logarithms

Topic 5-1a: Logarithms: The Inverse of an
Exponential

So what is the inverse of an exp onential function ?

It should be easy to de termine the inverse of each
of these functions:

f (x) = x + 3

g (x) = 7x

h(x) = x³

Yet, we don’t have anything previously covered that
serves as a proper inverse function to:

f (x) = 2^x

Definition: Let a be a positive real number other
than 1. Then the logarithm with base
a, denoted as, is defined as
fol lows :

Observations:
A logarithm with base a is the
inverse operation of an exponential
base a.

The output of a logarithmic
expression is an exponent .

Better said, a logarithmic expression
is an exponent.

Evaluate these basic logarithmic expressions by
considering their analogues amongst exponential
expressions:

Properties of Logarithms

Evaluate these basic logarithmic expressions by
applying the properties of logarithms and properties
of exponents:

Topic 5-1b: Logarithmic Functions & their
Transformations

Basic logarithmic functions are logically the inverse
functions to corresponding basic exponential
functions.

Compare the graphs of the exponential function
base two with the logarithmic function base two.

Definition:
For a logarithmic function
f (x) = log_a x, the domain is (0, ∞),
the range is (−∞, ∞), and a vertical
asymptote exists at x = 0.

Transformations are also possible with logarithmic
functions.

Ex. 1 Identify the basic function in the given
function.

Determine the transformations applied to the
basic function that produce the given
function. State the transformations applied,
in the correct order , using units and
directions as appropriate.

Finally, determine the domain, range, and
asymptote of the given function.

f (x) = −log₄(x + 2)

Ex. 2 Identify the basic function in the given
function.

Determine the transformations applied to the
basic function that produce the given
function. State the transformations applied,
in the correct order, using units and
directions as appropriate.

Finally, determine the domain, range, and
asymptote of the given function.

Logarithmic Functions based on limited information

Similar to exponential functions, it is possible to
graphically define a logarithmic function using
limited information. You need to know if the
function has been dilated, what its
y- intercept is , and one point other than the y-intercept.

Ex. 1 Find the logarithmic function of the form
f (x) = log_a x defined by the graph.

Ex. 2 Find the logarithmic function of the form
f (x) = log_a x defined by the graph.

Topic 5-1c: Special Logarithms

Common Logarithms : Base 10 logarithm
 

Applications include pH scale, Richter scale, decibel scale,
etc.

Natural Logarithms: Base e logarithm

Since the natural exponential function has so many uses, its
inverse should logically have many uses as well.

Properties of Special Logarithms

	General	Common	Natural
1		log 1= 0	ln 1= 0
2		log 10 = 1	ln e = 1
3
4

Natural Logarithms: Base e logarithm

Since the natural exponential function has so many uses, its
inverse should logically have many uses as well.