Solve 15^6-x=25

$$15 ^ { 6 - x } = 25$$

Solve for x

$$x=2\log_{15}\left(675\right)\approx 4.811367742$$

Steps Using Definition of Logarithm

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

$$15^{-x+6}=25$$

Take the logarithm of both sides of the equation.

$$\log(15^{-x+6})=\log(25)$$

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

$$\left(-x+6\right)\log(15)=\log(25)$$

Divide both sides by \log(15).

$$-x+6=\frac{\log(25)}{\log(15)}$$

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

$$-x+6=\log_{15}\left(25\right)$$

Subtract 6 from both sides of the equation.

$$-x=2\log_{15}\left(5\right)-6$$

Divide both sides by -1.

$$x=\frac{2\log_{15}\left(5\right)-6}{-1}$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(15)}+2\log_{15}\left(675\right)$$

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