Question
\( \operatorname { og } _ { 18 } ( 3 k - 2 ) = \log _ { 18 } 2 k \)
Ask by Weston Graham. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( k = 2 \)
Solution
It looks like there's a small typo in your equation. Assuming you meant to write \( \log \) instead of \( \operatorname{og} \), the equation is:
\[
\log_{18} (3k - 2) = \log_{18} (2k)
\]
To solve for \( k \), follow these steps:
1. **Equalize the Arguments:**
Since the logarithms are equal and have the same base (18), their arguments must be equal:
\[
3k - 2 = 2k
\]
2. **Solve for \( k \):**
Subtract \( 2k \) from both sides:
\[
3k - 2k - 2 = 0
\]
Simplify:
\[
k - 2 = 0 \quad \Rightarrow \quad k = 2
\]
3. **Verify the Solution:**
Ensure that the arguments of the logarithms are positive when \( k = 2 \):
- \( 3k - 2 = 3(2) - 2 = 6 - 2 = 4 > 0 \)
- \( 2k = 2(2) = 4 > 0 \)
Both arguments are positive, so \( k = 2 \) is a valid solution.
**Final Answer:**
\( k = 2 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve \( \operatorname{og}_{18} (3k - 2) = \log_{18} 2k \), we start by utilizing properties of logarithms. Since both sides have the same base, we can equate the arguments: \[ 3k - 2 = 2k. \] Now, let's isolate \( k \): \[ 3k - 2k = 2 \implies k = 2. \] This tells us that \( k \) must be equal to 2. To verify, we can substitute \( k \) back into the original equation: Left side: \[ \log_{18}(3(2) - 2) = \log_{18}(6 - 2) = \log_{18}(4). \] Right side: \[ \log_{18}(2(2)) = \log_{18}(4). \] Since both sides are equal, our solution \( k = 2 \) is correct.
