To solve the logarithmic equation [tex]\(\log (x-14)-\log (x-6)=\log 3\)[/tex], we'll follow a few steps.
### Step 1: Use Logarithm Properties
First, we use the logarithm property that states [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex].
So, we can rewrite the given equation as:
[tex]\[
\log \left(\frac{x-14}{x-6}\right) = \log 3
\][/tex]
### Step 2: Set the Arguments Equal
If [tex]\(\log A = \log B\)[/tex], then we have [tex]\(A = B\)[/tex].
Therefore, we set the arguments of the logarithms equal to each other:
[tex]\[
\frac{x-14}{x-6} = 3
\][/tex]
### Step 3: Solve the Rational Equation
Now we solve the rational equation for [tex]\(x\)[/tex]:
[tex]\[
\frac{x-14}{x-6} = 3
\][/tex]
To clear the denominator, we multiply both sides of the equation by [tex]\(x-6\)[/tex]:
[tex]\[
x-14 = 3(x-6)
\][/tex]
### Step 4: Distribute and Collect Terms
Next, we distribute the 3 on the right-hand side:
[tex]\[
x-14 = 3x - 18
\][/tex]
Then, we move all terms involving [tex]\(x\)[/tex] to one side and constants to the other side:
[tex]\[
x - 3x = -18 + 14
\][/tex]
Simplify the equation:
[tex]\[
-2x = -4
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{-4}{-2}
\][/tex]
Simplify:
[tex]\[
x = 2
\][/tex]
### Step 6: Check the Solution
Finally, we need to check if [tex]\(x = 2\)[/tex] is a valid solution by substituting it back into the original logarithmic expressions. Specifically, we need to make sure that the logarithms' arguments are positive:
[tex]\[
x-14 > 0 \quad \text{and} \quad x-6 > 0
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
2-14 = -12 \quad \text{(which is not positive)}
\][/tex]
[tex]\[
2-6 = -4 \quad \text{(which is not positive)}
\][/tex]
Since both [tex]\(2-14\)[/tex] and [tex]\(2-6\)[/tex] are negative, the logarithmic arguments are not valid.
### Conclusion
Therefore, [tex]\(x=2\)[/tex] does not satisfy the initial condition that the arguments of the logarithms must be positive numbers. Hence, there is no valid solution to the given logarithmic equation [tex]\(\log (x-14) - \log (x-6) = \log 3\)[/tex].
