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Chapter 10: Problem 318

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\frac{1}{4} e^{x}=3\)

Short Answer

Expert verified
The exact answer is \(x = \text{ln}(12)\) and the approximate answer is \(x \approx 2.485\).

Step by step solution

01

Isolate the exponential term

To solve the exponential equation \(\frac{1}{4} e^{x}=3\), the first step is to isolate the exponential term \(e^x\). To do this, multiply both sides of the equation by 4. This results in \(\frac{1}{4} e^{x} \times 4 = 3 \times 4\) which simplifies to \(e^{x} = 12\).
02

Apply the natural logarithm

Since the exponential term is now isolated, apply the natural logarithm (ln) to both sides of the equation to solve for \(x\). Thus, \(\text{ln}(e^x) = \text{ln}(12)\).
03

Simplify the logarithmic equation

Utilize the property of logarithms \(\text{ln}(e^x) = x\) to simplify the left side of the equation, yielding \(x = \text{ln}(12)\).
04

Calculate the exact and approximate answers

The exact answer is \(x = \text{ln}(12)\). To find the approximate value, use a calculator to evaluate \(\text{ln}(12)\). This gives \(x \approx 2.485\) when rounded to three decimal places.

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Key Concepts

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

Isolate the Exponential Term
To start solving any exponential equation like \(\frac{1}{4} e^{x}=3\), we need to isolate the exponential term first. This simplifies the problem and makes it easier to solve. In our exercise, the exponential term is \(e^x\). To isolate it, we multiply both sides of the equation by 4. This step ensures that \(e^x\) stands alone on one side of the equation.
Here's how:
\(\frac{1}{4} e^{x} \times 4 = 3 \times 4\)
This simplifies to:
\(e^{x} = 12\)
Now that our exponential term is isolated, we're ready to move to the next step!
Natural Logarithm
Applying the natural logarithm (ln) is a crucial next step for solving exponential equations. Taking the natural logarithm of both sides of our equation \(e^x = 12\) allows us to transform the equation from an exponential form to a linear form, which is easier to handle.
Natural logarithm comes from the properties of exponents and logarithms, which are inverses. The natural log (ln) of \(e^x\) simplifies to \(x\).
So, our equation transforms as follows:
\( \text{ln}(e^x) = \text{ln}(12)\)
Using the property that \( \text{ln}(e^x) = x\), we simplify this to:
\(x = \text{ln}(12)\)
The problem is now easy to solve with basic algebra skills!
Approximate Solutions
Once we have the exact solution \(x = \text{ln}(12)\), we often want to approximate it to a decimal value for easier understanding. To do this, you can use a calculator with a natural logarithm function.
Here's the process:
1. Input \(12\) and press the \(\text{ln}\) button on your calculator.
2. The display will show \( \text{ln}(12) \) which approximates to 2.485.
The approximate value of our solution, rounded to three decimal places, is \(\text{ln}(12) \approx 2.485\).
This makes it easier to interpret and use in practical applications!
Exact Solutions
Exact solutions are important because they give us a precise answer without any rounding error. In our problem, the exact solution is \( x = \text{ln}(12)\).
This form is the most accurate representation of the solution. Why? Because it's derived directly from the equation without any approximation.
Here鈥檚 a quick reminder of the steps to find the exact solution in our example:
1. Isolate the exponential term: \( e^x = 12 \)
2. Apply the natural logarithm: \( \text{ln}(e^x) = \text{ln}(12) \)
3. Simplify: \( x = \text{ln}(12) \)
Whenever possible, use the exact solution for further calculations to maintain precision!

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Most popular questions from this chapter

In the following exercises, solve each equation. \(3^{3 x+1}=81\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(3 \ln x+4 \ln y-2 \ln z\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{3} 36-\log _{3} 4\)

Explain the method you would use to solve these equations: \(3^{x+1}=81, \quad 3^{x+1}=75 .\) Does your method require logarithms for both equations? Why or why not?

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}\)
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