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# 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 â€” Exponential Equality

If bx = by, 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.

This property is a direct consequence of the fact that exponential functions are one-to-one functions. That is, if two exponential functions have the same output, bx and by, then their inputs, x and y, must be equal.

Example 1

Solve: 25 5x-8 = 625

 Solution First, write each expression using the same base, 5. On the left, use the Power of a Power Property. Use the Exponential Equality Property to set the exponents equal to each other. 25 5x - 8(52) 5x - 8 5 10x - 16 10x - 16 = 625= 54 = 54 = 4 Add 16 to both sides. Divide both sides by 10. 10xx = 20= 2

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

 Original equation: Is Is Is 25 5x - 8 25 5 Â· 2 - 8 252 625 = 625= 625 ? = 625 ? = 625 ? Yes

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 Â· 5 = 52

625 = 5 Â· 5 Â· 5 Â· 5 = 54

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

Example 2

Solve: 163x + 1 = 324x

 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 163x + 1 = 324x 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 2x = 8 but not to solve 2x = 9.

Recall that

Thus,