Solve 15^-10x=90

$$15 ^ { - 10 x } = 90$$

Solve for x

$$x=-\frac{\log_{15}\left(90\right)}{10}\approx -0.16616419$$

Steps Using Definition of Logarithm

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

$$15^{-10x}=90$$

Take the logarithm of both sides of the equation.

$$\log(15^{-10x})=\log(90)$$

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

$$-10x\log(15)=\log(90)$$

Divide both sides by \log(15).

$$-10x=\frac{\log(90)}{\log(15)}$$

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

$$-10x=\log_{15}\left(90\right)$$

Divide both sides by -10.

$$x=\frac{\log_{15}\left(90\right)}{-10}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{5\ln(15)}-\frac{\log_{15}\left(90\right)}{10}$$

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