Solve 100=50left(1+0.5right)^x

$$100=50 { \left(1+0.5 \right) }^{ x }$$

Solve for x

$$x=\log_{1.5}\left(2\right)\approx 1.709511291$$

Steps Using Definition of Logarithm

Divide both sides by 50.

$$\frac{100}{50}=\left(1+0.5\right)^{x}$$

Divide 100 by 50 to get 2.

$$2=\left(1+0.5\right)^{x}$$

Add 1 and 0.5 to get 1.5.

$$2=1.5^{x}$$

Swap sides so that all variable terms are on the left hand side.

$$1.5^{x}=2$$

Take the logarithm of both sides of the equation.

$$\log(1.5^{x})=\log(2)$$

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

$$x\log(1.5)=\log(2)$$

Divide both sides by \log(1.5).

$$x=\frac{\log(2)}{\log(1.5)}$$

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

$$x=\log_{1.5}\left(2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.5)}+\log_{1.5}\left(2\right)$$

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