Question
Solve exactly. \( e^{-51570 k}=\frac{1}{2} \) \( k=\square \) Entry tip: Write the answer in exact form using fractions or integers, not decimals. Your answer may also contain one or more logs or natural logs.
Ask by Robbins Sherman. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( k = \frac{\ln 2}{51570} \)
Solution
We start with the equation:
e^(–51570 k) = 1/2
Take the natural logarithm of both sides:
ln(e^(–51570 k)) = ln(1/2)
Using the property ln(e^x) = x, we get:
–51570 k = ln(1/2)
Recall that ln(1/2) = –ln2, so:
–51570 k = –ln2
Now divide both sides by –51570:
k = (–ln2)/(–51570)
Since the negatives cancel, the exact solution is:
k = ln2 / 51570
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Simplify this solution Beyond the Answer
To solve the equation \( e^{-51570 k}=\frac{1}{2} \), we can start by taking the natural logarithm of both sides. This gives: \[ -51570 k = \ln\left(\frac{1}{2}\right) \] Next, we can isolate \( k \): \[ k = -\frac{\ln\left(\frac{1}{2}\right)}{51570} \] Using the property of logarithms \(\ln\left(\frac{1}{2}\right) = -\ln(2)\), we can rewrite \( k \): \[ k = \frac{\ln(2)}{51570} \] So the exact solution is: \[ k = \frac{\ln(2)}{51570} \]
