Solve 3(13^x)=5

$$3 ( 13 ^ { x } ) = 5$$

Solve for x

$$x=\log_{13}\left(\frac{5}{3}\right)\approx 0.199156222$$

Steps Using Definition of Logarithm

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

$$3\times 13^{x}=5$$

Divide both sides by 3.

$$13^{x}=\frac{5}{3}$$

Take the logarithm of both sides of the equation.

$$\log(13^{x})=\log(\frac{5}{3})$$

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

$$x\log(13)=\log(\frac{5}{3})$$

Divide both sides by \log(13).

$$x=\frac{\log(\frac{5}{3})}{\log(13)}$$

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

$$x=\log_{13}\left(\frac{5}{3}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(13)}+\log_{13}\left(\frac{5}{3}\right)$$

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