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Chapter 4: Problem 11

Find a number \(t\) such that \(\ln (2 t+1)=-4\).

Short Answer

Expert verified
The short answer is: \(t \approx -0.148\).

Step by step solution

01

Remove the natural logarithm

To remove the natural logarithm from the equation, we will use the property mentioned in the analysis: if \(\ln a = b\), then \(a = e^b\). Applying this property to our equation, we get: \(2t + 1 = e^{-4}\)
02

Isolate \(t\)

Now, we need to isolate \(t\) in the equation. To do so, we will first subtract 1 from both sides of the equation: \(2t = e^{-4} - 1\) Next, we will divide both sides by 2: \(t = \frac{e^{-4} - 1}{2}\)
03

Compute the value of \(t\)

Using a calculator, we can now find the value of \(t\): \(t = \frac{e^{-4} - 1}{2} \approx -0.148\) So, the value of \(t\) that satisfies the given equation is approximately \(t \approx -0.148\).

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