Exponential Equations

Exponential Equations

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

Example 1

Solve the given equation. 2^(3x - 1) = 4

Make both side's bases the same.

23x - 1 = 22

Then both side's exponents are equal.

3x - 1 = 2
x = 1

Example 2

Solve the given equation. 3^(x - 5) = 9^4

Make both side's bases the same.

3x - 5 = 38

Power of a power

Then both side's exponents are equal.

x - 5 = 8
x = 13

Example 3

Solve the given equation. 7^(4x + 8) = 1

Make both side's bases the same.

74x + 8 = 70

Zero exponent

Then both side's exponents are equal.

4x + 8 = 0
x = -2

Example 4

Solve the given equation. 125*5^x = (1/25)^x

Make both side's bases the same.

53 + x = 5-2x

Right side
Negative exponent

Left side
Product of powers

Then both side's exponents are equal.

3 + x = -2x
x = -1

Example 5

Solve the given equation. 4^x = 3*2^x + 4

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 2x as a variable,
and solve it like a quadratic equation.

Solving a quadratic equation by factoring

2x = 4 = 22.
So x = 2.

2x = -1.
But 2x > 0.
So, in this case, there's no solution.

So x = 2.