To solve the equation [tex]\(\log (15 - 5x) - \log (3x - 2) - 2 = 0\)[/tex] for [tex]\(x\)[/tex], let's follow these steps:
1. Simplify the Logarithmic Expression:
Use the properties of logarithms to combine the logs:
[tex]\[
\log (15 - 5x) - \log (3x - 2) = \log \left( \frac{15 - 5x}{3x - 2} \right)
\][/tex]
Hence, the equation becomes:
[tex]\[
\log \left( \frac{15 - 5x}{3x - 2} \right) - 2 = 0
\][/tex]
2. Isolate the Logarithm:
Add 2 to both sides of the equation:
[tex]\[
\log \left( \frac{15 - 5x}{3x - 2} \right) = 2
\][/tex]
3. Convert Logarithmic Form to Exponential Form:
Recall that if [tex]\(\log_a (b) = c\)[/tex], then [tex]\(a^c = b\)[/tex]. Here, the logarithm is in base 10 by default:
[tex]\[
10^2 = \frac{15 - 5x}{3x - 2}
\][/tex]
Therefore:
[tex]\[
100 = \frac{15 - 5x}{3x - 2}
\][/tex]
4. Solve the Resulting Equation:
Multiply both sides by [tex]\((3x - 2)\)[/tex] to clear the fraction:
[tex]\[
100(3x - 2) = 15 - 5x
\][/tex]
Expand and combine like terms:
[tex]\[
300x - 200 = 15 - 5x
\][/tex]
Add [tex]\(5x\)[/tex] to both sides:
[tex]\[
300x + 5x - 200 = 15
\][/tex]
Combine like terms:
[tex]\[
305x - 200 = 15
\][/tex]
Add 200 to both sides:
[tex]\[
305x = 215
\][/tex]
Finally, divide both sides by 305:
[tex]\[
x = \frac{215}{305}
\][/tex]
5. Simplify the Fraction (if needed, though not necessary if the direct answer is known):
Reduce the fraction:
[tex]\[
x = \frac{43}{61}
\][/tex]
Given this detailed step-by-step solution, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\log (15 - 5x) - \log (3x - 2) - 2 = 0\)[/tex] is:
[tex]\[
x = \frac{43}{61}
\][/tex]
