Question
Solve for \( x \) \[ \log _{5}(3 x-7)=1 \]
Ask by Gonzalez Mccarthy. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x=4 \)
Solution
Solve the equation \( \log_{5}(3x-7)=1 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\log_{5}{\left(3x-7\right)}=1\)
- step1: Find the domain:
\(\log_{5}{\left(3x-7\right)}=1,x>\frac{7}{3}\)
- step2: Convert the logarithm into exponential form:
\(3x-7=5^{1}\)
- step3: Evaluate the power:
\(3x-7=5\)
- step4: Move the constant to the right side:
\(3x=5+7\)
- step5: Add the numbers:
\(3x=12\)
- step6: Divide both sides:
\(\frac{3x}{3}=\frac{12}{3}\)
- step7: Divide the numbers:
\(x=4\)
- step8: Check if the solution is in the defined range:
\(x=4,x>\frac{7}{3}\)
- step9: Find the intersection:
\(x=4\)
The solution to the equation \( \log_{5}(3x-7)=1 \) is \( x=4 \).
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Simplify this solution Beyond the Answer
To solve the equation \( \log_{5}(3x - 7) = 1 \), we start by rewriting the logarithmic equation in its exponential form. This means that: \[ 3x - 7 = 5^1 \] Which simplifies to: \[ 3x - 7 = 5 \] Next, we isolate \( 3x \) by adding 7 to both sides: \[ 3x = 5 + 7 \] \[ 3x = 12 \] Now, we can solve for \( x \) by dividing both sides by 3: \[ x = \frac{12}{3} \] \[ x = 4 \] Therefore, the solution is: \[ \boxed{4} \]
