Solve 1.01^x=1.1

$${ 1.01 }^{ x } = 1.1$$

Solve for x

$$x=\log_{1.01}\left(1.1\right)\approx 9.57859404$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$1.01^{x}=1.1$$

Take the logarithm of both sides of the equation.

$$\log(1.01^{x})=\log(1.1)$$

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

$$x\log(1.01)=\log(1.1)$$

Divide both sides by \log(1.01).

$$x=\frac{\log(1.1)}{\log(1.01)}$$

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

$$x=\log_{1.01}\left(1.1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.01)}+\log_{1.01}\left(1.1\right)$$

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