Solve by Factoring 56^(3x-8)=1

$56^{3 x - 8} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$56^{3 x - 8} - 1 = 0$

Step 2

Add $1$ to both sides of the equation.

$56^{3 x - 8} = 1$

Step 3

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

$\ln (56^{3 x - 8}) = \ln (1)$

Step 4

Expand $\ln (56^{3 x - 8})$ by moving $3 x - 8$ outside the logarithm.

$(3 x - 8) \ln (56) = \ln (1)$

Step 5

Simplify the left side.

Apply the distributive property.

$3 x \ln (56) - 8 \ln (56) = \ln (1)$

Step 6

Simplify the right side.

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

$3 x \ln (56) - 8 \ln (56) = 0$

Step 7

Add $8 \ln (56)$ to both sides of the equation.

$3 x \ln (56) = 8 \ln (56)$

Step 8

Divide each term in $3 x \ln (56) = 8 \ln (56)$ by $3 \ln (56)$ and simplify.

Divide each term in $3 x \ln (56) = 8 \ln (56)$ by $3 \ln (56)$ .

$\frac{3 x \ln (56)}{3 \ln (56)} = \frac{8 \ln (56)}{3 \ln (56)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (56)}{3 \ln (56)} = \frac{8 \ln (56)}{3 \ln (56)}$

Rewrite the expression.

$\frac{x \ln (56)}{\ln (56)} = \frac{8 \ln (56)}{3 \ln (56)}$

Cancel the common factor of $\ln (56)$ .

Cancel the common factor.

$\frac{x \ln (56)}{\ln (56)} = \frac{8 \ln (56)}{3 \ln (56)}$

Divide $x$ by $1$ .

$x = \frac{8 \ln (56)}{3 \ln (56)}$

Simplify the right side.

Cancel the common factor of $\ln (56)$ .

Cancel the common factor.

$x = \frac{8 \ln (56)}{3 \ln (56)}$

Rewrite the expression.

$x = \frac{8}{3}$