Question
If \( \ln (7 x+5)=2 \), then \( x=\square \)
Ask by Mathis Carrillo. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = \frac{e^{2}-5}{7} \)
Solution
Solve the equation \( \ln(7x+5)=2 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\ln{\left(7x+5\right)}=2\)
- step1: Find the domain:
\(\ln{\left(7x+5\right)}=2,x>-\frac{5}{7}\)
- step2: Convert the logarithm into exponential form:
\(7x+5=e^{2}\)
- step3: Move the constant to the right side:
\(7x=e^{2}-5\)
- step4: Divide both sides:
\(\frac{7x}{7}=\frac{e^{2}-5}{7}\)
- step5: Divide the numbers:
\(x=\frac{e^{2}-5}{7}\)
- step6: Check if the solution is in the defined range:
\(x=\frac{e^{2}-5}{7},x>-\frac{5}{7}\)
- step7: Find the intersection:
\(x=\frac{e^{2}-5}{7}\)
The solution to the equation \( \ln(7x+5)=2 \) is \( x = \frac{e^{2}-5}{7} \).
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Simplify this solution Extra Insights
To solve the equation \( \ln (7x + 5) = 2 \), we first exponentiate both sides to eliminate the natural logarithm. This gives us: \[ 7x + 5 = e^2 \] Next, we subtract 5 from both sides: \[ 7x = e^2 - 5 \] Now, we divide by 7 to isolate \( x \): \[ x = \frac{e^2 - 5}{7} \] Using an approximate value for \( e^2 \) (approximately 7.389), we can further compute: \[ x \approx \frac{7.389 - 5}{7} \approx \frac{2.389}{7} \approx 0.3413 \] Thus, the solution is \( x \approx 0.3413 \).
