Solve (2/5)^8x=(25/4)^2

$$( \frac { 2 } { 5 } ) ^ { 8 x } = ( \frac { 25 } { 4 } ) ^ { 2 }$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$\left(\frac{2}{5}\right)^{8x}=\frac{625}{16}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{5}\right)^{8x})=\log(\frac{625}{16})$$

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

$$8x\log(\frac{2}{5})=\log(\frac{625}{16})$$

Divide both sides by \log(\frac{2}{5}).

$$8x=\frac{\log(\frac{625}{16})}{\log(\frac{2}{5})}$$

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

$$8x=\log_{\frac{2}{5}}\left(\frac{625}{16}\right)$$

Divide both sides by 8.

$$x=-\frac{4}{8}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{4\ln(\frac{2}{5})}-\frac{\log_{\frac{2}{5}}\left(\frac{4}{25}\right)}{4}$$

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