Solve 115=1.002^x

$$115 = 1.002 ^ { x }$$

Solve for x

$$x=\log_{1.002}\left(115\right)\approx 2374.837740214$$

Steps Using Definition of Logarithm

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

$$1.002^{x}=115$$

Take the logarithm of both sides of the equation.

$$\log(1.002^{x})=\log(115)$$

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

$$x\log(1.002)=\log(115)$$

Divide both sides by \log(1.002).

$$x=\frac{\log(115)}{\log(1.002)}$$

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

$$x=\log_{1.002}\left(115\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.002)}+\log_{1.002}\left(115\right)$$

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