Solve 5^(x+2)=8.5

$$5 ^ { ( x + 2 ) } = 8.5$$

Solve for x

$$x=-\log_{5}\left(\frac{2}{17}\right)-2\approx -0.67030213$$

Steps Using Definition of Logarithm

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

$$5^{x+2}=8.5$$

Take the logarithm of both sides of the equation.

$$\log(5^{x+2})=\log(8.5)$$

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

$$\left(x+2\right)\log(5)=\log(8.5)$$

Divide both sides by \log(5).

$$x+2=\frac{\log(8.5)}{\log(5)}$$

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

$$x+2=\log_{5}\left(8.5\right)$$

Subtract 2 from both sides of the equation.

$$x=\frac{\ln(\frac{17}{2})}{\ln(5)}-2$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5)}-\log_{5}\left(\frac{2}{17}\right)-2$$

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