Solve 5^5+x=52

$${ 5 }^{ 5+x } =52$$

Solve for x

$$x=\log_{5}\left(52\right)-5\approx -2.544954243$$

Steps Using Definition of Logarithm

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

$$5^{x+5}=52$$

Take the logarithm of both sides of the equation.

$$\log(5^{x+5})=\log(52)$$

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

$$\left(x+5\right)\log(5)=\log(52)$$

Divide both sides by \log(5).

$$x+5=\frac{\log(52)}{\log(5)}$$

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

$$x+5=\log_{5}\left(52\right)$$

Subtract 5 from both sides of the equation.

$$x=\log_{5}\left(52\right)-5$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5)}+\log_{5}\left(52\right)-5$$

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