Solving Exponential Equations

An exponential equation is an equation where one or more of the exponents contains a variable. Some types of exponential equations can be solved using the following property.

Property — Exponential Equality

If b^x = b^y, then x = y. Here, b > 0 and b ≠ 1.

Note:

Recall that for a one-to-one function, two different inputs always result in two different outputs.

That is, in a one-to-one function, each output value corresponds to exactly one input value.

Example 1

Solve: 25 ^5x-8 = 625

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We can check the solution by replacing x with 2 in the original equation and simplifying.

So, the solution of 25 ^{5x - 8} = 625 is x = 2.

Note:

To write each base in exponential form, first find its prime factorization.

For example:

25 = 5 · 5 = 5²

625 = 5 · 5 · 5 · 5 = 5⁴

The Power of a Power Property says (xm)n 5 xmn.

Example 2

Solve: 16^{3x + 1} = 32^4x

Solution Write each expression with the same base, 2. Use the Power of a Power Property. Use the Exponential Equality Property to set the exponents equal to each other.	163x + 1 (24)3x + 1 212x + 4 12x + 4	= 324x = (25)4x = 220x = 20x
Subtract 12x from both sides.	4	= 8x
Divide both sides by 8.		= x

We can check the solution in the usual way.

So, the solution of 16^{3x + 1} = 32^4x is

Note:

This method can only be used if we write each side of the equation as an exponential expression using the same base. If not, we must use a different method.

For example, we can use this method solve 2^x = 8 but not to solve 2^x = 9.

Recall that

Thus,