Solve 11=1.1^x

$$11= { 1.1 }^{ x }$$

Solve for x

$$x=\log_{1.1}\left(11\right)\approx 25.158857928$$

Steps Using Definition of Logarithm

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

$$1.1^{x}=11$$

Take the logarithm of both sides of the equation.

$$\log(1.1^{x})=\log(11)$$

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

$$x\log(1.1)=\log(11)$$

Divide both sides by \log(1.1).

$$x=\frac{\log(11)}{\log(1.1)}$$

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

$$x=\log_{1.1}\left(11\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.1)}+\log_{1.1}\left(11\right)$$

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