Solve 41.5=69.3^23x

$$41.5 = 69.3 ^ { 23 x }$$

Solve for x

$$x=\frac{\log_{69.3}\left(41.5\right)}{23}\approx 0.038218421$$

Steps Using Definition of Logarithm

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

$$69.3^{23x}=41.5$$

Take the logarithm of both sides of the equation.

$$\log(69.3^{23x})=\log(41.5)$$

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

$$23x\log(69.3)=\log(41.5)$$

Divide both sides by \log(69.3).

$$23x=\frac{\log(41.5)}{\log(69.3)}$$

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

$$23x=\log_{69.3}\left(41.5\right)$$

Divide both sides by 23.

$$x=\frac{\ln(\frac{83}{2})}{23\ln(\frac{693}{10})}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{23\ln(69.3)}-\frac{\log_{69.3}\left(\frac{2}{83}\right)}{23}$$

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