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Chapter 4: Problem 36

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{4 x-5}-7=11,243$$

Short Answer

Expert verified
The solution to the exponential equation \(e^{4x-5} = 11,243\) is \(x \approx 3.20\).

Step by step solution

01

Simplify the equation

First, add 7 to both sides of the equation to isolate the exponential term. This gives: \(e^{4x-5} = 11,250\).
02

Apply natural logarithm

In order to isolate \(x\), take the natural logarithm (ln) on both sides. This yields: \(ln(e^{4x-5}) = ln(11250)\). The left side simplifies using the property of logarithms \(ln(e^{a}) = a\), resulting in: \(4x - 5 = ln(11250)\).
03

Solve for x

Next, solve for \(x\) by first adding 5 to both sides and then dividing by 4. This leads to the solution: \(x = (ln(11250) + 5) / 4\).
04

Find decimal approximation

Finally, using a calculator, approximate this solution to two decimal places: \(x \approx 3.20\).

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