Solve 15^x=10^5

$$15 ^ { x } = 10 ^ { 5 }$$

Solve for x

$$x=5\log_{15}\left(10\right)\approx 4.251370769$$

Steps Using Definition of Logarithm

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

$$15^{x}=100000$$

Take the logarithm of both sides of the equation.

$$\log(15^{x})=\log(100000)$$

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

$$x\log(15)=\log(100000)$$

Divide both sides by \log(15).

$$x=\frac{\log(100000)}{\log(15)}$$

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

$$x=\log_{15}\left(100000\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(15)}+5\log_{15}\left(10\right)$$

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