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Chapter 4: Problem 117

Find the approximate solution to each equation. Round to four decimal places. $$5 e^{x}=4$$

Short Answer

Expert verified
The approximate solution is \(x \approx -0.2231\).

Step by step solution

01

Isolate the exponential term

Start by isolating the exponential term. In the given equation, divide both sides by 5:\[e^{x} = \frac{4}{5}\]
02

Apply the natural logarithm

Apply the natural logarithm (\(\ln\)) to both sides to get rid of the exponential function:\[\ln(e^{x}) = \ln\left(\frac{4}{5}\right)\]
03

Simplify using logarithm properties

Use the property of logarithms, \(\ln(e^{x}) = x\), to simplify the left side:\[x = \ln\left(\frac{4}{5}\right)\]
04

Calculate the logarithm

Calculate the value of \(\ln\left(\frac{4}{5}\right)\) using a calculator:\[x \approx \ln(0.8) \approx -0.2231\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

isolate exponential term
The first step in solving an exponential equation like \(5 e^{x} = 4\) is to isolate the exponential term. Here, the exponential term is \(e^{x}\). To isolate it, divide both sides of the equation by 5:

\[e^{x} = \frac{4}{5}\]

This simplifies the problem, allowing us to focus solely on the exponential part. With the term now isolated, we can move on to the next steps needed to solve for \(x\).
natural logarithm
To eliminate the exponential term in the equation \(e^{x} = \frac{4}{5}\), we use the natural logarithm.

The natural logarithm, denoted as \(\text{ln}\), is especially useful when working with the base \(e\). Applying the natural logarithm to both sides of the isolated equation gives us:

\[\text{ln}(e^{x}) = \text{ln}\bigg(\frac{4}{5}\bigg)\]

This step makes the equation easier to handle because of the properties of logarithms.
properties of logarithms
Logarithms have several useful properties that simplify complex equations. One key property is that the natural logarithm of an exponential function, \( \text{ln}(e^{x}) \), simplifies to \( x \).

Applying this property to our equation \(\text{ln}(e^{x}) = \text{ln}\bigg(\frac{4}{5}\bigg)\):

\[ x = \text{ln}\bigg(\frac{4}{5}\bigg) \]

With this conversion, we have a straightforward equation to solve. This property essentially 'cancels out' the exponential nature, leaving just \( x \).
calculate logarithm
Finally, we calculate \(\text{ln}\bigg(\frac{4}{5}\bigg)\) to find the value of \( x \). Using a calculator or logarithm table:

\[ x = \text{ln}(0.8) \]

Approximating the natural logarithm:

\[ x \approx \text{ln}(0.8) \approx -0.2231 \]

Thus, the solution to the equation \( 5 e^{x} = 4 \) is \( x \approx -0.2231 \). Remember to always round to the required precision, in this case, four decimal places.

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