Solve 8^x-2=1+4^5

$${ 8 }^{ x-2 } =1+ { 4 }^{ 5 }$$

Solve for x

$$x = \frac{\log_{2} {(1025)} + 6}{3} \approx 5.333802731$$

Steps Using Definition of Logarithm

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

$$8^{x-2}=1025$$

Take the logarithm of both sides of the equation.

$$\log(8^{x-2})=\log(1025)$$

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

$$\left(x-2\right)\log(8)=\log(1025)$$

Divide both sides by \log(8).

$$x-2=\frac{\log(1025)}{\log(8)}$$

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

$$x-2=\log_{8}\left(1025\right)$$

Add 2 to both sides of the equation.

$$x=\frac{\log_{2}\left(1025\right)}{3}-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(2)}+\frac{\log_{2}\left(1025\right)}{3}+2$$

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