How to Solve Exponential Equations Using Logarithms

What is the technique to solve exponential equations involving logarithms?

To solve exponential equations involving logarithms, we can use the property of logarithms that allows us to take the logarithm of both sides of the equation. This helps in isolating the variable in the exponent and solving for its value. Let's take an example equation and break down the steps to solve it.

Let's consider the equation 5^(x+8) = 7. To solve this equation, we follow these steps:

Step 1: Take the logarithm of both sides

Using the property of logarithms, we take the logarithm of both sides of the equation:

log₁₀(5^(x+8)) = log₁₀7

Step 2: Apply the laws of logarithms

We can simplify the equation using the laws of logarithms:

(x + 8)log₁₀5 = log₁₀7

Step 3: Solve for the variable

Divide both sides by log₁₀5 to isolate x:

x + 8 = log₁₀7/log₁₀5

x = (log₅7) - 8

By following these steps and understanding the properties of logarithms, we can efficiently solve exponential equations like the given example.