Question

What is the exact solution to the equation 4^5x = 3^(x-2)? SHOW ALL WORK.

Answer 1

The exact solution to the equation 4^5x = 3^(x-2) is found by applying the natural logarithm to both sides, using the power rule of logarithms, and isolating x which results in x = -2 · ln(3) / (5 · ln(4) - ln(3)).

Step-by-step explanation:

To solve the equation 4^5x = 3^(x-2), we need to apply logarithms because the equation involves variables within exponential functions. Here's how we do it step by step:

1. First, take the natural logarithm (ln) of both sides of the equation ln(4^5x) = ln(3^(x-2)).
2. Next, use the power rule of logarithms to bring down the exponents: 5x · ln(4) = (x-2) · ln(3).
3. Now, we'll distribute the logarithms to get 5x · ln(4) = x · ln(3) - 2 · ln(3).
4. To solve for x, we get all the terms containing x on one side: 5x · ln(4) - x · ln(3) = -2 · ln(3).
5. Factor x out from the left side: x(5 · ln(4) - ln(3)) = -2 · ln(3).
6. Finally, divide both sides by (5 · ln(4) - ln(3)) to isolate x, resulting in x = -2 · ln(3) / (5 · ln(4) - ln(3)).

Therefore, the exact solution to the equation is x = -2 · ln(3) / (5 · ln(4) - ln(3)).