Solve -10+32*2^-0.2x=0

$$-10+32 \times { 2 }^{ -0.2x } =0$$

Solve for x

$$x=-5\log_{2}\left(5\right)+20\approx 8.390359526$$

Steps Using Definition of Logarithm

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

$$32\times 2^{-0.2x}-10=0$$

Add 10 to both sides of the equation.

$$32\times 2^{-0.2x}=10$$

Divide both sides by 32.

$$2^{-0.2x}=\frac{5}{16}$$

Take the logarithm of both sides of the equation.

$$\log(2^{-0.2x})=\log(\frac{5}{16})$$

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

$$-0.2x\log(2)=\log(\frac{5}{16})$$

Divide both sides by \log(2).

$$-0.2x=\frac{\log(\frac{5}{16})}{\log(2)}$$

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

$$-0.2x=\log_{2}\left(\frac{5}{16}\right)$$

Multiply both sides by -5.

$$x=\frac{\log_{2}\left(5\right)-4}{-0.2}$$

Solve for x (complex solution)

$$x=-\frac{i\times 10\pi n_{1}}{\ln(2)}-5\log_{2}\left(5\right)+20$$

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