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Chapter 4: Problem 16

Find a number \(y\) such that \(e^{4 y-3}=5\).

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: \[y \approx \frac{\ln(5) + 3}{4} \approx 1.2041\]

Step by step solution

01

Take the natural logarithm of both sides

We will take the natural logarithm of both sides of the equation to get rid of the exponent. \[\ln\left(e^{(4y-3)}\right) = \ln(5)\]
02

Use the logarithmic property

Recall that the logarithmic property states that \(\ln(a^b)=b\ln(a)\). We will use this property to bring down the exponent: \[(4y-3)\ln(e) = \ln(5)\] Since the natural logarithm of e is equal to 1 (\(\ln(e)=1\)), the left side of the equation simplifies to: \[4y - 3 = \ln(5)\]
03

Isolate y

Now we have to isolate \(y\). First, we add 3 to both sides of the equation: \[4y = \ln(5) + 3\] Then, we divide both sides of the equation by 4: \[y = \frac{\ln(5) + 3}{4}\]
04

Find the numerical value of y

Using a calculator, we can now find the numerical value of \(y\): \[y \approx \frac{\ln(5) + 3}{4} \approx 1.2041\] Therefore, the value of \(y\) that makes the equation \(e^{4y-3}=5\) true is approximately \(1.2041\).

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