High School

Solve the equation [tex]\log _3 x - \log _3 (x-4) = 2[/tex]

A. [tex]x = 3.5[/tex]

B. [tex]x = 4.5[/tex]

C. [tex]x = 5.5[/tex]

D. [tex]x = 6.5[/tex]

E. [tex]x = 7.5[/tex]

Answer :

To solve the equation [tex]\(\log_3 x - \log_3(x-4) = 2\)[/tex], we'll follow these steps:

1. Use Logarithm Properties: The equation involves logarithms with the same base, so we can use the property:
[tex]\[
\log_b a - \log_b c = \log_b \left( \frac{a}{c} \right)
\][/tex]
Applying this property to [tex]\(\log_3 x - \log_3(x-4)\)[/tex], we get:
[tex]\[
\log_3 \left( \frac{x}{x-4} \right) = 2
\][/tex]

2. Convert the Logarithmic Equation to Exponential Form: The equation [tex]\(\log_3 \left( \frac{x}{x-4} \right) = 2\)[/tex] can be rewritten in exponential form:
[tex]\[
\frac{x}{x-4} = 3^2
\][/tex]
Simplifying, we have:
[tex]\[
\frac{x}{x-4} = 9
\][/tex]

3. Solve the Equation: Multiply both sides by [tex]\(x - 4\)[/tex] to eliminate the fraction:
[tex]\[
x = 9(x - 4)
\][/tex]

Expand the right-hand side:
[tex]\[
x = 9x - 36
\][/tex]

Rearrange the terms:
[tex]\[
x - 9x = -36
\][/tex]

Simplify:
[tex]\[
-8x = -36
\][/tex]

Divide both sides by -8:
[tex]\[
x = \frac{36}{8} = \frac{9}{2}
\][/tex]

This simplifies to:
[tex]\[
x = 4.5
\][/tex]

4. Verify the Solution: We want to ensure the solution is valid. Notice that the original logarithmic expressions are only defined for values of [tex]\(x\)[/tex] where both [tex]\(x\)[/tex] and [tex]\(x-4\)[/tex] are greater than zero. Thus:
[tex]\[
x > 4
\][/tex]

Since [tex]\(x = 4.5\)[/tex] meets this condition, it is a valid solution.

Therefore, the solution to the logarithmic equation is [tex]\(\boxed{4.5}\)[/tex].