Solve for x 64^(7x-8)=1

$64^{7 x - 8} = 1$

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

$\ln (64^{7 x - 8}) = \ln (1)$

Expand $\ln (64^{7 x - 8})$ by moving $7 x - 8$ outside the logarithm.

$(7 x - 8) \ln (64) = \ln (1)$

Simplify the left side.

Apply the distributive property.

$7 x \ln (64) - 8 \ln (64) = \ln (1)$

Simplify the right side.

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

$7 x \ln (64) - 8 \ln (64) = 0$

Add $8 \ln (64)$ to both sides of the equation.

$7 x \ln (64) = 8 \ln (64)$

Divide each term in $7 x \ln (64) = 8 \ln (64)$ by $7 \ln (64)$ and simplify.

Divide each term in $7 x \ln (64) = 8 \ln (64)$ by $7 \ln (64)$ .

$\frac{7 x \ln (64)}{7 \ln (64)} = \frac{8 \ln (64)}{7 \ln (64)}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x \ln (64)}{7 \ln (64)} = \frac{8 \ln (64)}{7 \ln (64)}$

Rewrite the expression.

$\frac{x \ln (64)}{\ln (64)} = \frac{8 \ln (64)}{7 \ln (64)}$

Cancel the common factor of $\ln (64)$ .

Cancel the common factor.

$\frac{x \ln (64)}{\ln (64)} = \frac{8 \ln (64)}{7 \ln (64)}$

Divide $x$ by $1$ .

$x = \frac{8 \ln (64)}{7 \ln (64)}$

Simplify the right side.

Cancel the common factor of $\ln (64)$ .

Cancel the common factor.

$x = \frac{8 \ln (64)}{7 \ln (64)}$

Rewrite the expression.

$x = \frac{8}{7}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{7}$

Decimal Form:

$x = 1.\overline{142857}$

Mixed Number Form:

$x = 1 \frac{1}{7}$