Evaluate 4^(3x-2)=1

$4^{3 x - 2} = 1$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4^{3 x - 2}) = \ln (1)$

Step 2

Expand $\ln (4^{3 x - 2})$ by moving $3 x - 2$ outside the logarithm.

$(3 x - 2) \ln (4) = \ln (1)$

Step 3

Simplify the left side.

Apply the distributive property.

$3 x \ln (4) - 2 \ln (4) = \ln (1)$

Step 4

Simplify the right side.

The natural logarithm of $1$ is $0$ .

$3 x \ln (4) - 2 \ln (4) = 0$

Step 5

Add $2 \ln (4)$ to both sides of the equation.

$3 x \ln (4) = 2 \ln (4)$

Step 6

Divide each term in $3 x \ln (4) = 2 \ln (4)$ by $3 \ln (4)$ and simplify.

Divide each term in $3 x \ln (4) = 2 \ln (4)$ by $3 \ln (4)$ .

$\frac{3 x \ln (4)}{3 \ln (4)} = \frac{2 \ln (4)}{3 \ln (4)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (4)}{3 \ln (4)} = \frac{2 \ln (4)}{3 \ln (4)}$

Rewrite the expression.

$\frac{x \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{3 \ln (4)}$

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{3 \ln (4)}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (4)}{3 \ln (4)}$

Simplify the right side.

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$x = \frac{2 \ln (4)}{3 \ln (4)}$

Rewrite the expression.

$x = \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}$

Decimal Form:

$x = 0.\overline{6}$