Question

Maria wrote the equation log(x/2)+log(20/x^2)=log8. What is the solution to Maria's equation?

Answer 1

Answer:

C.

Step-by-step explanation:

Answer 2

The solution is x=5/4.

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

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

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

Exponentiate:
$10^0=\frac{10}{8x} \\ \\1=\frac{10}{8x}$

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