Solve 68=2.5^x

$$68= { 2.5 }^{ x }$$

Solve for x

$$x=\log_{2.5}\left(68\right)\approx 4.604987869$$

Steps Using Definition of Logarithm

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

$$2.5^{x}=68$$

Take the logarithm of both sides of the equation.

$$\log(2.5^{x})=\log(68)$$

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

$$x\log(2.5)=\log(68)$$

Divide both sides by \log(2.5).

$$x=\frac{\log(68)}{\log(2.5)}$$

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

$$x=\log_{2.5}\left(68\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2.5)}+\log_{2.5}\left(68\right)$$

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