Solve 5^2x+3=3.5

$${ 5 }^{ 2x+3 } =3.5$$

Solve for x

$$x=\frac{\log_{5}\left(3.5\right)-3}{2}\approx -1.110807301$$

Steps Using Definition of Logarithm

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

$$5^{2x+3}=3.5$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x+3})=\log(3.5)$$

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

$$\left(2x+3\right)\log(5)=\log(3.5)$$

Divide both sides by \log(5).

$$2x+3=\frac{\log(3.5)}{\log(5)}$$

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

$$2x+3=\log_{5}\left(3.5\right)$$

Subtract 3 from both sides of the equation.

$$2x=\frac{\ln(\frac{7}{2})}{\ln(5)}-3$$

Divide both sides by 2.

$$x=\frac{\frac{\ln(\frac{7}{2})}{\ln(5)}-3}{2}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(5)}+\frac{\log_{5}\left(3.5\right)}{2}-1.5$$

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