Exponential Equation

To solve an exponential equation,
make the bases of both sides the same.

The base of the left side 2^{3x - 1} is 2.

So change the base of the right side to 2:
4 = 2².

The bases of both sides are the same.
Then the exponents of both sides are the same.

So 3x - 1 = 2.

Solve 3x - 1 = 2.

Move -1 to the right side.

Divide both sides by 3.
Then x = 1.

So x = 1.

The base of the left side 3^{x - 5} is 3.
So change the base of the right side to 3.

9 = 3²

(3²)⁴ = 3^2⋅4 = 3⁸

Power of a Power

3^{x - 5} = 3⁸

The bases of both sides are the same.
Then the exponents of both sides are the same.

So x - 5 = 8.

Solve x - 5 = 8.

Move -5 to the right side.
Then x = 13.

So x = 13.

The base of the left side 7^{4x + 8} is 7.
So change the base of the right side to 7.

1 = 7⁰

Zero Exponent

The bases of both sides are the same.
Then the exponents of both sides are the same.

So 4x + 8 = 0.

Solve 4x + 8 = 0.

Move +8 to the right side.

Divide both sides by 4.
Then x = -2.

So x = -2.

The bases of the numbers are 125, 5, and 1/25.
So change the bases of the numbers to 5.

125 = 5³

(1/25)^x = 25^-x

Negative Exponent

5³⋅5³ = 5^{3 + x}

Product of Powers

25^-x = (5²)^-x

(5²)^-x = 5^-2x

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

The bases of both sides are the same.
Then the exponents of both sides are the same.

So 3 + x = -2x.

Solve 3 + x = -2x.

Move 3 to the right side.
Move -2x to the left side.

Then, x + 2x, 3x is equal to -3.

Divide both sides by 3.
Then x = -1.

So x = -1.

The given equation has 3 terms,
not 2 terms.

So you can't directly compare the exponents
like the previous examples.

To solve this equation,
first move the right side terms to the left side.

4^x = (2²)^x = 2^2x

2^2x = (2^x)²

Think 2^x as a variable
and factor (2^x)² x - 3⋅2^x - 4 = 0.

Find a pair of numbers
whose product is the constant term -4
and whose sum is the coefficient of the middle term -3.

-4⋅1 = -4
-4 + 1 = -3

So (2^x - 4)(2^x + 1) = 0.

Factor a Quadratic Trinomial

Solve the quadratic equation for each case.

Case 1: 2^x - 4 = 0

Move -4 to the right side.
Then 2^x = 4.

Change the base of 4 to 2.
4 = 2²

2^x = 2²

So x = 2.

This is the answer for case 1.

Case 2: 2^x + 1 = 0

Move +1 to the right side.
Then 2^x = -1.

See 2^x = -1.

The left side 2^x is always plus.
(The graph of y = a^x is always above the x-axis.)

But the right side -1 is minus.

So this equation has no solution.

This is the answer for case 2.

For case 1, x = 2.
For case 2, there's no solution.

So the solution of (2^x - 4)(2^x + 1) = 0 is
x = 2.

So x = 2 is the answer.