Solve 5^2x=1.5

$${ 5 }^{ 2x } =1.5$$

Solve for x

$$x=\frac{\log_{5}\left(1.5\right)}{2}\approx 0.125964818$$

Steps Using Definition of Logarithm

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

$$5^{2x}=1.5$$

Take the logarithm of both sides of the equation.

$$\log(5^{2x})=\log(1.5)$$

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

$$2x\log(5)=\log(1.5)$$

Divide both sides by \log(5).

$$2x=\frac{\log(1.5)}{\log(5)}$$

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

$$2x=\log_{5}\left(1.5\right)$$

Divide both sides by 2.

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

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(5)}-\frac{\log_{5}\left(\frac{2}{3}\right)}{2}$$

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