Solve f(x)=200*1.16^x=1000

$$f ( x ) = 200 \cdot 1.16 ^ { x } = 1000$$

Solve for x

$$x=\log_{1.16}\left(5\right)\approx 10.843807148$$

Steps Using Definition of Logarithm

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

$$200\times 1.16^{x}=1000$$

Divide both sides by 200.

$$1.16^{x}=5$$

Take the logarithm of both sides of the equation.

$$\log(1.16^{x})=\log(5)$$

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

$$x\log(1.16)=\log(5)$$

Divide both sides by \log(1.16).

$$x=\frac{\log(5)}{\log(1.16)}$$

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

$$x=\log_{1.16}\left(5\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.16)}+\log_{1.16}\left(5\right)$$

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