Logarithmic Functions

In this lesson, we will look at how to evaluate simple logarithmic functions and solve for x in logarithmic functions.

Solving for x in Logarithmic Equations

Example:

Calculate the value of each of the following:
a) 1og ₂ 64
b) log ₉ 3
c) log ₄ 1
d) log ₆ 6
e) log ₈ 0.25
f) log ₃–9

Solution:

a) Let x = log ₂ 64
2^x = 64
x = 6

b) Let x = log ₉ 3
9^x = 3
x =

c) Let x = log ₄ 1
4^x = 1
x = 0

d) Let x = log ₆ 6
6^x = 6
x = 1

e) Let x = log ₈ 0.25
8^x = 0.25

f) Let x = log ₃– 9
3^x = – 9

Since it is not possible for 3^x to be negative, log ₃–9 is undefined.

Example

Solve log_x 4 = 2

Solution:

log_x 4 = 2
x² = 4
x = 2 or –2

Since x is the base, x > 0 and x ≠ 1; so x = –2 is rejected and the only solution is x = 2

Example:

Solve log ₃ x = 2

Solution:

log ₃ x = 2
3² = x
x = 9

Example:

Solve log _x (4x – 3) = 2

Solution:

log _x (4x – 3) = 2
x² = 4x – 3
x² – 4x + 3 = 0
(x -1)(x – 3) = 0
So, x = 1 or 3

For the logarithm to be defined, the only solution is 3.

Take note of the following:

• Logarithms of a number to the base of the same number is 1, i.e. log_a a = 1
• Loarithms of 1 to any base is 0, i.e. log_a 1 = 0
• Log_a 0 is undefined
• Logarithms of negative numbers are undefined.
• The base of logarithms cannot be nagative or 1.

Videos

Finding the value of a logarithmic function -
Professor Edward Burger explains finding the value of a logarithmic function.

Solving for x in logarithmic equations -
Professor Edward Burger explains solving for x in logarithmic equations.

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