Solve 30/23={left(101/100right)}^x

$$\frac{ 30 }{ 23 } = { \left( \frac{ 101 }{ 100 } \right) }^{ x }$$

Solve for x

$$x=\log_{\frac{101}{100}}\left(\frac{30}{23}\right)\approx 26.702947837$$

Steps Using Definition of Logarithm

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

$$\left(\frac{101}{100}\right)^{x}=\frac{30}{23}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{101}{100}\right)^{x})=\log(\frac{30}{23})$$

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

$$x\log(\frac{101}{100})=\log(\frac{30}{23})$$

Divide both sides by \log(\frac{101}{100}).

$$x=\frac{\log(\frac{30}{23})}{\log(\frac{101}{100})}$$

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

$$x=\log_{\frac{101}{100}}\left(\frac{30}{23}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(\frac{101}{100})}+\log_{\frac{101}{100}}\left(\frac{30}{23}\right)$$

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