Precalculus Examples

$2 \ln (x - 1) = \ln (4 x)$

Step 1

Divide both sides by $2$ to remove it from the front of the logarithm.

$\frac{2 \ln (x - 1)}{2} = \frac{\ln (4 x)}{2}$

Step 2

Simplify each side of the equation.

$\ln (x - 1) = \ln (2 x^{\frac{1}{2}})$

Step 3

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = e$ , $x = x - 1$ , and $y = \ln (2 x^{\frac{1}{2}})$ .

$b = e$

$x = x - 1$

$y = \ln (2 x^{\frac{1}{2}})$

Step 4

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$e^{\ln (2 x^{\frac{1}{2}})} = x - 1$