Solve 1.12^n=2

$$1.12 ^ { n } = 2$$

Solve for n

$$n=\log_{1.12}\left(2\right)\approx 6.116255374$$

Steps Using Definition of Logarithm

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

$$1.12^{n}=2$$

Take the logarithm of both sides of the equation.

$$\log(1.12^{n})=\log(2)$$

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

$$n\log(1.12)=\log(2)$$

Divide both sides by \log(1.12).

$$n=\frac{\log(2)}{\log(1.12)}$$

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

$$n=\log_{1.12}\left(2\right)$$

Solve for n (complex solution)

$$n=\frac{i\times 2\pi n_{1}}{\ln(1.12)}+\log_{1.12}\left(2\right)$$

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