Solving Logarithmic Equations Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

Solve

\log_3(x+6) - \log_3(x-2) = 2

Solution

1
Apply the quotient rule: $\log_3\left(\frac{x+6}{x-2}\right) = 2$ . Convert to exponential form: $\frac{x+6}{x-2} = 3^2 = 9$ .
2
Solve: $x + 6 = 9(x-2) = 9x - 18$ . So $24 = 8x$ , giving $x = 3$ . Check: $\log_3(9) - \log_3(1) = 2 - 0 = 2$ . ✓

Answer

x = 3

The quotient rule combines two logarithms into one, then converting to exponential form removes the logarithm entirely. Checking the solution confirms