Solve 139^n=199

$$139 ^ { n } = 199$$

Solve for n

$$n=\log_{139}\left(199\right)\approx 1.072719179$$

Steps Using Definition of Logarithm

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

$$139^{n}=199$$

Take the logarithm of both sides of the equation.

$$\log(139^{n})=\log(199)$$

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

$$n\log(139)=\log(199)$$

Divide both sides by \log(139).

$$n=\frac{\log(199)}{\log(139)}$$

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

$$n=\log_{139}\left(199\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(139)}+\log_{139}\left(199\right)$$

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