# Exponential Equations

How to solve exponential equations problems: examples and their solutions.

## Example 1

Make both side's bases the same.

2^{3x - 1} = 2^{2}

Then both side's exponents are equal.

3*x* - 1 = 2*x* = 1

## Example 2

Make both side's bases the same.

3^{x - 5} = 3^{8}

Then both side's exponents are equal.*x* - 5 = 8*x* = 13

## Example 3

Make both side's bases the same.

7^{4x + 8} = 7^{0}

Zero exponent

Then both side's exponents are equal.

4*x* + 8 = 0*x* = -2

## Example 4

Make both side's bases the same.

5^{3 + x} = 5^{-2x}

Right side

Negative exponent

Left side

Product of powers

Then both side's exponents are equal.

3 + *x* = -2*x**x* = -1

## Example 5

This equation cannot be solved

by simply setting bases the same.

+4 is bothering.

And 3 is not a cooperative number

to change the base to 2.

To solve this example,

think 2^{x} as a variable,

and solve it like a quadratic equation.

Solving a quadratic equation by factoring

2^{x} = 4 = 2^{2}.

So *x* = 2.

2^{x} = -1.

But 2^{x} > 0.

So, in this case, there's no solution.

So *x* = 2.