What are two different ways that you can solve the exponential equation?

There are at least two common methods for solving exponential equations: rewriting the equation with the same base on both sides, and taking the logarithm of both sides.

Method 1: Rewriting with the Same Base

This method relies on the property of equality for exponential functions. If you can express both sides of the equation using the same base, you can then equate the exponents and solve for the variable.

Example:

Solve for x: 2^x = 8

1. Rewrite 8 as a power of 2: 8 = 2³
2. Substitute: 2^x = 2³
3. Equate the exponents: x = 3

Therefore, the solution is x = 3. This works because the exponential function is one-to-one.

Method 2: Taking the Logarithm of Both Sides

When you cannot easily rewrite both sides of the equation with the same base, you can take the logarithm of both sides. This method utilizes the power rule of logarithms, which allows you to bring the exponent down as a coefficient. You can use any base for the logarithm, but common choices are base 10 (log) or base e (ln).

Example:

Solve for x: 5^x = 17

1. Take the logarithm of both sides (using base 10): log(5^x) = log(17)
2. Apply the power rule of logarithms: x * log(5) = log(17)
3. Solve for x: x = log(17) / log(5)
4. Approximate using a calculator: x ≈ 1.760

Therefore, the solution is x ≈ 1.760. Natural logarithms (ln) would work as well, yielding the same answer: x = ln(17) / ln(5) ≈ 1.760.