To solve the equation [tex]\(\log [7(x-7)]=\log [3(2 x)]\)[/tex], let's follow a systematic approach.
1. Understanding the Equation:
We have two logarithmic expressions set equal to each other:
[tex]\[
\log [7(x - 7)] = \log [3(2x)]
\][/tex]
2. Eliminate the Logarithms:
Since [tex]\(\log a = \log b\)[/tex] implies [tex]\(a = b\)[/tex] (logarithms with the same base and arguments), we can set the arguments equal to each other:
[tex]\[
7(x - 7) = 3(2x)
\][/tex]
3. Simplify the Equation:
Expand both sides of the equation:
[tex]\[
7x - 49 = 6x
\][/tex]
4. Isolate the Variable:
To isolate [tex]\(x\)[/tex], subtract [tex]\(6x\)[/tex] from both sides:
[tex]\[
7x - 6x - 49 = 0
\][/tex]
Simplify:
[tex]\[
x - 49 = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Add 49 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 49
\][/tex]
6. Check the Solution:
Substitute [tex]\(x = 49\)[/tex] back into the original arguments to ensure both sides are equal:
- For the left side:
[tex]\[
7(49 - 7) = 7 \cdot 42 = 294
\][/tex]
- For the right side:
[tex]\[
3(2 \cdot 49) = 3 \cdot 98 = 294
\][/tex]
Both sides are equal, confirming the solution is correct.
Thus, the solution to the equation is:
[tex]\[
x = 49
\][/tex]
