7.4 Evaluate Logarithms and Graph Logarithmic Functions

Logarithm of y with base b

A logarithm denoted by log_b y and defined as log_b y = x if and only if b^x = y, given that b and y are positive numbers with b ¹ 1.

Common logarithm

A logarithm with base 10

Natural logarithm

A logarithm with base e

DEFINITION OF LOGARITHM WITH BASE b

Let b and y be positive numbers with b ¹ 1. The logarithm of y with base b is denoted by log_b y is defined as follows:

log_b y = _x_ if and only if b^x = _y_

The expression log_b y is read as "log base b of y."

Example 1

Rewrite logarithmic equations

Logarithmic Form	Exponential Form
a. log2 32 = 5	_2^5 = 32_
b. log7 1 = 0	_7^0 = 1_
c. log13 13 = 1	_13^1 = 13_
d. log1/2 2 = -1	_____= 2_

Example 2

Evaluate the logarithm.

a. log3 81	b. log4 0.25
c. log1/4 256	d. log49 7

Solution

To help you find the value of log_b y, ask yourself what power of b gives you y.

a. 3 to what power gives you 81?

3__4__ = 81, so log₃ 81 = __4__.

b. 4 to what power gives you 0.25?

4^__-1__ = 0.25, so log₄ 0.25 = __-1__.

c. to what power gives you 256?

^__-4__ = 256, so log_1/4 256 = _-4_.

d. 49 to what power gives you 7?

49^_1/2_ = 7, so log₄₉ 7 = _{___ .}

Example 3

Use inverse properties

Simplify the expression.

a. 10^log 6.7

b. log2 16^x

Solution

a. 10^log 6.7 = _6.7_	logb b^x = x
b. log2 16^x = _log2(2^4)^x_	Express 16 as a power with base _2_.
= _log2 2^4x_	Power of a power property
= _4x_	logb b^x = x

Example 4

Find the inverse of the function

a. y = log_3/2 x b. y = In(x - 4)

a. From the definition of logarithm, the inverse of y = log_3/2 x is y =

b. y = In(x - 4)	Write original function.
_x = In(y - 4)_	Switch x and y.
_e^x = y - 4_	Write in exponential form.
_e^x + 4_ = y	Solve for y.
The inverse of y = In(x - 4) is y = _e^x + 4_ .

PARENT GRAPHS FOR LOGARITHMIC FUNCTIONS

The graph of y = log_b x is shown below for b > 1 and for 0 < b < 1. Because y = log_b x and y = b^x are __inverse__ functions, the graph of y = log_b x is the reflection of the graph of y = b^x in the line __y = x__.

Note that the y-axis is a vertical asymptote of the graph of y = log_b x. The domain of y = log_b x is _x > 0_ , and the range is __all real numbers__ .

Example 5

Graph (a) y = log₂ x and (b) y = log_1/3 x.

a. Plot several convenient points, such as (1, _0_ ), (2, _1_ ) , and (4, _2_ ). The y-axis is a _vertical asymptote_. From left to right, draw a curve that starts just to the _right_ of the y-axis and moves _up_ through the plotted points.

b. Plot several convenient points, such as (1, _0_ ), (3, _-1_ ), and (9, _-2_ ). The y-axis is a _vertical asymptote_. From left to right, draw a curve that starts just to the _right_ of the y-axis and moves _down_ through the plotted points.

Example 6

Translate a logarithmic graph

Graph y = log₃(x - 1) + 2. State the domain and range.

Sketch the graph of the parent function y = log₃ x, which passes through (1, _0_),
(3, _1_), and (9, _2_).

Translate the parent graph _right 1 unit_ and _up 2 units_. The translated graph passes through (2, _2_), (4, _3_), and (10, _4_). The graph's asymptote is _x = 1_. The domain is _x > 1_, and the range is _all real numbers_.