Solve 105^n=700

$$105 ^ { n } = 700$$

Solve for n

$$n=\log_{105}\left(700\right)\approx 1.407635614$$

Steps Using Definition of Logarithm

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

$$105^{n}=700$$

Take the logarithm of both sides of the equation.

$$\log(105^{n})=\log(700)$$

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

$$n\log(105)=\log(700)$$

Divide both sides by \log(105).

$$n=\frac{\log(700)}{\log(105)}$$

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

$$n=\log_{105}\left(700\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(105)}+\log_{105}\left(700\right)$$

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