To solve the equation [tex]\(\log_4(x) - \log_4(x - 1) = \frac{1}{2}\)[/tex], let's proceed step-by-step.
### Step 1: Use the properties of logarithms
Recall the logarithmic property: [tex]\(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\)[/tex].
Applying this property:
[tex]\[
\log_4(x) - \log_4(x - 1) = \log_4\left(\frac{x}{x-1}\right)
\][/tex]
Therefore, the given equation becomes:
[tex]\[
\log_4\left(\frac{x}{x-1}\right) = \frac{1}{2}
\][/tex]
### Step 2: Convert the logarithmic equation to an exponential equation
By definition of logarithms, if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(b^c = a\)[/tex].
We convert the equation [tex]\(\log_4\left(\frac{x}{x-1}\right) = \frac{1}{2}\)[/tex] to its exponential form:
[tex]\[
4^{1/2} = \frac{x}{x-1}
\][/tex]
### Step 3: Simplify the exponential equation
Recall that [tex]\(4^{1/2} = \sqrt{4} = 2\)[/tex]:
[tex]\[
2 = \frac{x}{x-1}
\][/tex]
### Step 4: Solve the resulting equation for [tex]\(x\)[/tex]
Set up the equation:
[tex]\[
2 = \frac{x}{x-1}
\][/tex]
To clear the fraction, multiply both sides by [tex]\(x - 1\)[/tex]:
[tex]\[
2(x - 1) = x
\][/tex]
Distribute and simplify:
[tex]\[
2x - 2 = x
\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[
2x - x - 2 = 0
\][/tex]
Combine like terms:
[tex]\[
x - 2 = 0
\][/tex]
Add 2 to both sides:
[tex]\[
x = 2
\][/tex]
### Step 5: Verify the solution
It is always important to check that the solution satisfies the original equation.
Substitute [tex]\(x = 2\)[/tex] back into the original equation:
[tex]\[
\log_4(2) - \log_4(1) = \frac{1}{2}
\][/tex]
Recall that [tex]\(\log_b(1) = 0\)[/tex] for any base [tex]\(b\)[/tex]:
[tex]\[
\log_4(2) - 0 = \frac{1}{2}
\][/tex]
Simplifying, we have:
[tex]\[
\log_4(2) = \frac{1}{2}
\][/tex]
This is true since [tex]\(4^{1/2} = 2\)[/tex].
Therefore, the solution [tex]\(x = 2\)[/tex] satisfies the original equation.
### Conclusion
The solution to the equation [tex]\(\log_4(x) - \log_4(x - 1) = \frac{1}{2}\)[/tex] is:
[tex]\[
x = 2
\][/tex]
