Solving Logarithmic Equations Math Example 3

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

Example 3

medium

Solve

\ln(x^2 - 4) = \ln(3x)

Solution

1
Since $\ln(A) = \ln(B)$ implies $A = B$ (when both are positive): $x^2 - 4 = 3x$ , so $x^2 - 3x - 4 = 0$ . Factor: $(x-4)(x+1) = 0$ .
2
Check: $x = 4$ : $\ln(12) = \ln(12)$ ✓. $x = -1$ : $\ln(-3)$ is undefined ✗. Only $x = 4$ .

Answer

x = 4

When both sides of an equation are single logarithms with the same base, the arguments must be equal. However, you must verify that the solutions make all logarithm arguments positive. Here