Problem 12 - Thomas Calculus

Question

In Exercises 11 and \(12,\) solve for \(k\). a. \(e^{5 k}=\frac{1}{4} \quad\) b. \(80 e^{k}=1 \quad\) c. \(e^{(\ln 0.8) k}=0.8\)

Step-by-Step Solution

Answer

a. \(k = \frac{\ln\left(\frac{1}{4}\right)}{5}\) b. \(k = \ln\left(\frac{1}{80}\right)\) c. \(k = 1\)

Step 1: Converting to Logarithms

For equation (a), we have: \[ e^{5k} = \frac{1}{4} \]To solve for \(k\), let's take the natural logarithm on both sides, which gives:\[ \ln(e^{5k}) = \ln\left(\frac{1}{4}\right) \]Using the logarithm power rule, this simplifies to:\[ 5k = \ln\left(\frac{1}{4}\right) \]

Step 2: Solving for k in Equation (a)

Now, divide both sides by 5 to solve for \(k\):\[ k = \frac{\ln\left(\frac{1}{4}\right)}{5} \]

Step 3: Solving for k in Equation (b)

For equation (b), we start with:\[ 80e^{k} = 1 \]Divide both sides by 80:\[ e^{k} = \frac{1}{80} \]Take the natural logarithm on both sides:\[ \ln(e^{k}) = \ln\left(\frac{1}{80}\right) \]This simplifies to:\[ k = \ln\left(\frac{1}{80}\right) \]

Step 4: Handling the Exponent in Equation (c)

For equation (c), start with:\[ e^{(\ln 0.8)k} = 0.8 \]Recognize that the exponent can be simplified:\[ (\ln 0.8)k = \ln(0.8) \]Divide both sides by \(\ln(0.8)\):\[ k = \frac{\ln(0.8)}{\ln(0.8)} \]

Step 5: Simplifying Equation (c)

From the previous step, the expression simplifies to:\[ k = 1 \]

Key Concepts

Solving Logarithms

Logarithms are a helpful tool in solving equations that involve exponents, especially when variables are in the power. By converting exponential equations to logarithmic form, we can easily isolate the variable and solve the equation.
For example, if we have an equation like \( e^{5k} = \frac{1}{4} \), it can often be challenging to tackle this directly. However, using logarithms simplifies this task.
Here's how it works:

• Identify the exponent equation, such as \( e^{5k} \).
• Take the logarithm on both sides of the equation. When dealing with base \( e \), the natural logarithm \( \ln \) is typically employed.
• Use properties of logarithms to bring the exponent down, making the equation linear in terms of \( k \). For instance, \( \ln(e^{5k}) = \ln(\frac{1}{4}) \) simplifies by applying the logarithm power rule to \( 5k = \ln(\frac{1}{4}) \).
• After that, solve the linear equation for \( k \) by isolating it.

With these steps, you can effectively solve logarithmic problems, making them much less daunting.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a specific logarithm with a base of \( e \), where \( e \) is approximately 2.718. Natural log is extensively used in mathematical calculations, particularly involving exponential growth and decay.
Here's how to work with natural logarithms:

• When solving equations like \( 80e^{k} = 1 \), the natural log helps to eliminate the exponent. By applying \( \ln \) on both sides, the exponent \( k \) can be brought down: \( \ln(e^{k}) = \ln(\frac{1}{80}) \).
• The natural log has unique characteristics, such as \( \ln(e) = 1 \) and \( \ln(1) = 0 \), which simplify exponential expressions when \( e \) is part of the basis.
• Understanding how to manipulate natural logs is crucial when dealing with equations in exponential form.

Mastering the use of \( \ln \) can make it easier to handle equations, especially in calculus and real-world applications like calculating interest compounding.

Logarithm Power Rule

The logarithm power rule is a fundamental property that allows converting the exponent of an expression into a coefficient, making it more manageable to solve. It states that \( \ln(a^b) = b \cdot \ln(a) \).
Utilizing the logarithm power rule often simplifies calculations significantly:

• In the problem \( e^{5k} = \frac{1}{4} \), applying the power rule results in \( \ln(e^{5k}) = 5k \cdot \ln(e) \). Given that \( \ln(e) = 1 \), the expression further simplifies to \( 5k = \ln(\frac{1}{4}) \).
• This simplification is advantageous as it decreases the complexity by reducing the power into a linear form.
• The rule applies to any logarithmic base, not just the natural log, broadening its usability.

Thus, the logarithm power rule is a powerful technique, especially in tackling complex exponential equations, providing a more straightforward path to the solution.