Solve 100(1+0.50)^n=500

$$100 ( 1 + 0.50 ) ^ { n } = 500$$

Solve for n

$$n=\log_{1.5}\left(5\right)\approx 3.969362296$$

Steps Using Definition of Logarithm

Divide both sides by 100.

$$\left(1+0.5\right)^{n}=\frac{500}{100}$$

Divide 500 by 100 to get 5.

$$\left(1+0.5\right)^{n}=5$$

Add 1 and 0.5 to get 1.5.

$$1.5^{n}=5$$

Take the logarithm of both sides of the equation.

$$\log(1.5^{n})=\log(5)$$

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

$$n\log(1.5)=\log(5)$$

Divide both sides by \log(1.5).

$$n=\frac{\log(5)}{\log(1.5)}$$

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

$$n=\log_{1.5}\left(5\right)$$