SOLUTION: Solve the following exponential functions a. 5 (with an exponent of) 5x+1 = 125 b. 3 (with an exponent of x squared) = 81 (with an exponent of x)

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Question 380913: Solve the following exponential functions
a. 5 (with an exponent of) 5x+1 = 125
b. 3 (with an exponent of x squared) = 81 (with an exponent of x)

Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
With many equations where the variable is in an exponent, one usually uses logarithms to solve the equation. An exception is when both sides of the equation can be expressed as powers of the same number. Both of your equations fit into this later category.

a.
Since we can write this as:

Both sides are not powers of the same number, 5. In order for these powers of 5 to be equal, the exponents must be equal:
5x+1 = 3
This is a simple equation to solve:
5x = 2
x = 2/5

P.S. As you noted in your thank-you note, I overlooked the exponent on the 81 in the second problem. Here's the corrected solution:
b.
Since we can rewrite this as:

Using the power of a power rule for exponents (i.e. multiply) we get:

Again this equation is true only if the exponents are equal:

This is a quadratic equation. To solve it we need one side equal to zero. Subtracting 4x from each side we get:

Now we factor (or use the Quadratic Formula). This factors very simply:
x(x-4) = 0
From the zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
x = 0 or x-4 = 0
Solving the second equation we get:
x = 0 or x = 4

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