Solve 100+3.2^x=200

$$100+ { 3.2 }^{ x } =200$$

Solve for x

$$x=2\log_{3.2}\left(10\right)\approx 3.959220204$$

Steps Using Definition of Logarithm

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

$$3.2^{x}+100=200$$

Subtract 100 from both sides of the equation.

$$3.2^{x}=100$$

Take the logarithm of both sides of the equation.

$$\log(3.2^{x})=\log(100)$$

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

$$x\log(3.2)=\log(100)$$

Divide both sides by \log(3.2).

$$x=\frac{\log(100)}{\log(3.2)}$$

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

$$x=\log_{3.2}\left(100\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(3.2)}+2\log_{3.2}\left(10\right)$$

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