Solve 5{10}^frac{x{4}}=32

$$5 { 10 }^{ \frac{ x }{ 4 } } = 32$$

Solve for x

$$x=-4\log_{10}\left(\frac{5}{32}\right)\approx 3.224719896$$

Steps Using Definition of Logarithm

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

$$5\times 10^{\frac{1}{4}x}=32$$

Divide both sides by 5.

$$10^{\frac{1}{4}x}=\frac{32}{5}$$

Take the logarithm of both sides of the equation.

$$\log(10^{\frac{1}{4}x})=\log(\frac{32}{5})$$

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

$$\frac{1}{4}x\log(10)=\log(\frac{32}{5})$$

Multiply both sides by 4.

$$x=\frac{\log(\frac{32}{5})}{\frac{1}{4}}$$