Maria wrote the equation log(x/2)+log(20/x^2)=log8. What is the solution to Maria's equation?a. x=3/10b. x=4/5c. x=5/4d. x10/3

The solution is x=5/4.

We use the properties of logs to rewrite the equation:
[tex]\log[(\frac{x}{2})(\frac{20}{x^2})]=\log8 \\ \\\log(\frac{20x}{2x^2})=\log8 \\ \\\log(\frac{10}{x})=\log8[/tex]

Get all of the logs on the same side of the equation y subtracting log 8:
[tex]\log(\frac{10}{x})-\log8=\log8 - \log8 \\ \\\log(\frac{10}{x})-\log8=0[/tex]

Use the properties of logs to rewrite:
[tex]\log(\frac{10}{x}\div8)=0 \\ \\\log(\frac{10}{x}\div\frac{8}{1})=0 \\ \\\log(\frac{10}{x}\times\frac{1}{8})=0 \\ \\\log(\frac{10}{8x})=0[/tex]

Exponentiate:
[tex]10^0=\frac{10}{8x} \\ \\1=\frac{10}{8x}[/tex]

Multiply both sides by 8x:
1*8x = (10/8x)*8x
8x=10

Divide both sides by 8:
8x/8 = 10/8
x = 10/8 = 5/4