Solve (12/13)^4*(13/12)^-8=(12/13)^2x

$$( \frac { 12 } { 13 } ) ^ { 4 } \times ( \frac { 13 } { 12 } ) ^ { - 8 } = ( \frac { 12 } { 13 } ) ^ { 2 x }$$

Solve for x

$$x=6$$

Steps Using Definition of Logarithm

Calculate \frac{12}{13} to the power of 4 and get \frac{20736}{28561}.

$$\frac{20736}{28561}\times \left(\frac{13}{12}\right)^{-8}=\left(\frac{12}{13}\right)^{2x}$$

Calculate \frac{13}{12} to the power of -8 and get \frac{429981696}{815730721}.

$$\frac{20736}{28561}\times \frac{429981696}{815730721}=\left(\frac{12}{13}\right)^{2x}$$

Multiply \frac{20736}{28561} and \frac{429981696}{815730721} to get \frac{8916100448256}{23298085122481}.

$$\frac{8916100448256}{23298085122481}=\left(\frac{12}{13}\right)^{2x}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{12}{13}\right)^{2x}=\frac{8916100448256}{23298085122481}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{12}{13}\right)^{2x})=\log(\frac{8916100448256}{23298085122481})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$2x\log(\frac{12}{13})=\log(\frac{8916100448256}{23298085122481})$$

Divide both sides by \log(\frac{12}{13}).

$$2x=\frac{\log(\frac{8916100448256}{23298085122481})}{\log(\frac{12}{13})}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$2x=\log_{\frac{12}{13}}\left(\frac{8916100448256}{23298085122481}\right)$$

Divide both sides by 2.

$$x=\frac{12}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(\frac{12}{13})}-\log_{\frac{12}{13}}\left(\frac{4826809}{2985984}\right)$$

$$n_{1}\in \mathrm{Z}$$