Solve 5^y-9=15

$$5 ^ { y - 9 } = 15$$

Solve for y

$$y=\log_{5}\left(3\right)+10\approx 10.682606194$$

Steps Using Definition of Logarithm

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

$$5^{y-9}=15$$

Take the logarithm of both sides of the equation.

$$\log(5^{y-9})=\log(15)$$

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

$$\left(y-9\right)\log(5)=\log(15)$$

Divide both sides by \log(5).

$$y-9=\frac{\log(15)}{\log(5)}$$

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

$$y-9=\log_{5}\left(15\right)$$

Add 9 to both sides of the equation.

$$y=\log_{5}\left(15\right)-\left(-9\right)$$

Solve for y (complex solution)

$$y=\frac{2\pi n_{1}i}{\ln(5)}+\log_{5}\left(3\right)+10$$

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