Logarithmic Equations

Logarithmic Equations

How to solve logarithmic equations: examples, and their solutions.

Example 1

Solve the given equation. log_3 x = 4

(gray) should be (+).
So write x > 0.

Solve the log.

Logarithmic form

x = 81 satisfies x > 0.

So x = 81 is the answer.

Example 2

Solve the given equation. log_x 64 = 3

The base (brown) should be
0 < (base) < 1 or (base) > 1.

So write
0 < x < 1 or x > 1.

Solve the log.

Logarithmic form

x = 64 satisfies 0 < x < 1 or x > 1.

So x = 64 is the answer.

Example 3

Solve the given equation. log_2 (log_3 (log_5 x)) = 0

(gray) should be (+).
So write x > 0.

Solve the log.

Logarithmic form

Solve the log.

Solve the log.

x = 125 satisfies x > 0.

So x = 125 is the answer.

Example 4

Solve the given equation. log_2 x + log_2 (x - 1) = log_2 12

(gray) should be (+).
So x > 0, x - 1 > 0.

The intersection is x > 1.

Logarithms of products

Both logs' bases are the same: 2.

So x(x - 1) = 12.

Solving a quadratic equation by factoring.

x = 4, -3.

But -3 doesn't satisfy x > 1.

So x = 4 is the answer.

Example 5

Solve the given equation. (log_5 x)^2 - log_5 x^2 - 3 = 0

(gray) should be (+).
So x > 0, x2 > 0.

The intersection is x > 0.

Logarithms of powers

Think 'log5 x' as a variable
and solve the quadratic equation.

Solving a quadratic equation by factoring.

x = 125, 1/5 satisfy x > 0.

So x = 125, 1/5 are the answer.