Certainly! Let’s solve the given logarithmic equation step-by-step:
We start with the logarithmic equation:
[tex]\[ \log_4(x) - \log_4(x - 9) = \frac{1}{2} \][/tex]
1. Apply the Quotient Rule for Logarithms:
The quotient rule states that [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex]. Applying this rule to our equation, we get:
[tex]\[ \log_4\left(\frac{x}{x - 9}\right) = \frac{1}{2} \][/tex]
2. Convert to Exponential Form:
To convert the logarithmic equation to its exponential form, we use the fact that [tex]\(\log_b(a) = c \)[/tex] is equivalent to [tex]\( b^c = a\)[/tex]. Thus,
[tex]\[ 4^{\frac{1}{2}} = \frac{x}{x - 9} \][/tex]
3. Simplify the Exponential Expression:
[tex]\(4^{\frac{1}{2}}\)[/tex] is the square root of 4.
[tex]\[ \sqrt{4} = 2 \][/tex]
So our equation now becomes:
[tex]\[ 2 = \frac{x}{x - 9} \][/tex]
4. Solve for [tex]\(x\)[/tex]:
We can eliminate the denominator by cross-multiplying:
[tex]\[ 2(x - 9) = x \][/tex]
[tex]\[ 2x - 18 = x \][/tex]
5. Isolate [tex]\(x\)[/tex]:
Subtract [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[ 2x - x = 18 \][/tex]
[tex]\[ x = 18 \][/tex]
Therefore, the solution to the equation is:
[tex]\[ x \approx 18 \][/tex]
