Solve 50000=32000left(1+0.05right)^x

$$50000 = 32000 { \left( 1+0.05 \right) }^{ x }$$

Solve for x

$$x=2\log_{1.05}\left(1.25\right)\approx 9.147071141$$

Steps Using Definition of Logarithm

Divide both sides by 32000.

$$\frac{50000}{32000}=\left(1+0.05\right)^{x}$$

Reduce the fraction \frac{50000}{32000} to lowest terms by extracting and canceling out 2000.

$$\frac{25}{16}=\left(1+0.05\right)^{x}$$

Add 1 and 0.05 to get 1.05.

$$\frac{25}{16}=1.05^{x}$$

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

$$1.05^{x}=\frac{25}{16}$$

Take the logarithm of both sides of the equation.

$$\log(1.05^{x})=\log(\frac{25}{16})$$

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

$$x\log(1.05)=\log(\frac{25}{16})$$

Divide both sides by \log(1.05).

$$x=\frac{\log(\frac{25}{16})}{\log(1.05)}$$

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

$$x=\log_{1.05}\left(\frac{25}{16}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.05)}+2\log_{1.05}\left(1.25\right)$$

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