Solve 2^0.1x-5=7

$$2 ^ { 0.1 x } - 5 = 7$$

Solve for x

$$x=10\left(\log_{2}\left(3\right)+2\right)\approx 35.849625007$$

Steps Using Definition of Logarithm

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

$$2^{0.1x}-5=7$$

Add 5 to both sides of the equation.

$$2^{0.1x}=12$$

Take the logarithm of both sides of the equation.

$$\log(2^{0.1x})=\log(12)$$

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

$$0.1x\log(2)=\log(12)$$

Divide both sides by \log(2).

$$0.1x=\frac{\log(12)}{\log(2)}$$

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

$$0.1x=\log_{2}\left(12\right)$$

Multiply both sides by 10.

$$x=\frac{\log_{2}\left(12\right)}{0.1}$$

Solve for x (complex solution)

$$x=\frac{i\times 20\pi n_{1}}{\ln(2)}+10\log_{2}\left(3\right)+20$$

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