Solve 3800=123^x

$$3800 = 123 ^ { x }$$

Solve for x

$$x=\log_{123}\left(3800\right)\approx 1.712892885$$

Steps Using Definition of Logarithm

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

$$123^{x}=3800$$

Take the logarithm of both sides of the equation.

$$\log(123^{x})=\log(3800)$$

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

$$x\log(123)=\log(3800)$$

Divide both sides by \log(123).

$$x=\frac{\log(3800)}{\log(123)}$$

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

$$x=\log_{123}\left(3800\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(123)}+\log_{123}\left(3800\right)$$

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