Question
If \( \log _{3}(7 x-2)=5 \), then what is the value of \( x ? \)
Ask by Pearson Bates. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 35 \)
Solution
To solve the equation \( \log_{3}(7x - 2) = 5 \), we'll use the definition of logarithms:
\[
\log_{b}(A) = C \quad \text{means} \quad b^C = A
\]
Applying this to the given equation:
\[
\log_{3}(7x - 2) = 5 \quad \Rightarrow \quad 3^5 = 7x - 2
\]
Calculate \( 3^5 \):
\[
3^5 = 243
\]
So,
\[
243 = 7x - 2
\]
Add 2 to both sides:
\[
245 = 7x
\]
Finally, divide both sides by 7 to solve for \( x \):
\[
x = \frac{245}{7} = 35
\]
**Answer:** 35
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve the equation \( \log _{3}(7x-2)=5 \), we first need to rewrite it in its exponential form. This gives us: \[ 7x - 2 = 3^5 \] Calculating \( 3^5 \): \[ 3^5 = 243 \] Now we can substitute this back into the equation: \[ 7x - 2 = 243 \] Adding 2 to both sides: \[ 7x = 245 \] Now, divide by 7: \[ x = \frac{245}{7} = 35 \] Thus, the value of \( x \) is \( \boxed{35} \).
