# Logarithmic Equations

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

## Example 1

(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

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

(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

(gray) should be (+).

So *x* > 0, *x* - 1 > 0.

The intersection is *x* > 1.

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

(gray) should be (+).

So *x* > 0, *x*^{2} > 0.

The intersection is *x* > 0.

Think 'log_{5} *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.