Solve 2400/1200=(109/100)^x

$$\frac { 2400 } { 1200 } = ( \frac { 109 } { 100 } ) ^ { x }$$

Solve for x

$$x=\log_{\frac{109}{100}}\left(2\right)\approx 8.043231727$$

Steps Using Definition of Logarithm

Divide 2400 by 1200 to get 2.

$$2=\left(\frac{109}{100}\right)^{x}$$

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

$$\left(\frac{109}{100}\right)^{x}=2$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{109}{100}\right)^{x})=\log(2)$$

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

$$x\log(\frac{109}{100})=\log(2)$$

Divide both sides by \log(\frac{109}{100}).

$$x=\frac{\log(2)}{\log(\frac{109}{100})}$$

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

$$x=\log_{\frac{109}{100}}\left(2\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{109}{100})}+\log_{\frac{109}{100}}\left(2\right)$$

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