Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 37-40, use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take ln of both sides, obtaining \(x \ln 3=\ln 11\); then \(x=(\ln 11) /(\ln 3) \approx 2.1827\). \(5^{x}=13\)

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 1.5933\).

Step 1: Take the natural logarithm of both sides

To solve the equation \(5^x = 13\), we start by taking the natural logarithm of both sides. This gives us: \(\ln(5^x) = \ln(13)\).

Step 2: Apply the power rule of logarithms

The power rule of logarithms states that \(\ln(a^b) = b \cdot \ln(a)\). Applying this rule to the left side of the equation, we get: \(x \cdot \ln(5) = \ln(13)\).

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation by \(\ln(5)\): \(x = \frac{\ln(13)}{\ln(5)}\).

Step 4: Calculate the value

Now, compute the values of the natural logarithms and divide: \(x \approx \frac{2.5649}{1.6094} \approx 1.5933\). Therefore, the approximate solution to the equation is \(x \approx 1.5933\).

Key Concepts

Natural Logarithms

Natural logarithms play a crucial role in solving exponential equations. They are logarithms with the base of the mathematical constant \(e\), approximately equal to 2.718. In the context of solving equations like \(5^x = 13\), natural logarithms help by providing a way to "bring down" the exponent. This process simplifies the equation by converting the exponent into a coefficient.When you take the natural logarithm of both sides of an equation, you're essentially using a powerful tool to manage complex exponential expressions. Remember:

• The natural logarithm of an exponential expression like \(b^x\) is \(\ln(b^x)\).
• This transforms the equation from an exponential form to a more manageable linear form.

In practice, this means applying \(\ln\) to both sides, which nicely sets the stage for using the power rule of logarithms. Furthermore, natural logarithms are incredibly useful in both theoretical mathematics and practical engineering problems, thanks to their relationship with \(e\).

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept that makes dealing with exponents significantly easier. It states that for any positive \(a\), and real number \(b\), the expression \(\ln(a^b)\) simplifies to \(b \cdot \ln(a)\). This is a key tool in solving equations involving exponents because it transforms a potentially tricky power into a multiplication.When solving an equation like \(5^x = 13\), applying the power rule allows you to bring the exponent "\(x\)" down, making it a coefficient that is easier to manipulate:

• Start by writing your expression with natural logarithms: \(\ln(5^x)\).
• Apply the power rule: \(x \cdot \ln(5)\).

This transformation is crucial because it takes a complex exponential problem and turns it into a simple linear equation. Once this step is achieved, the equation can be solved using basic algebraic techniques, making the power rule of logarithms an invaluable asset in your mathematical toolkit.

Solving Equations

Solving equations using logarithms hinges on understanding the relationship between exponents and logarithms. More specifically, it involves translating exponential forms into logarithmic ones so that simpler algebraic operations can be used.For the equation \(5^x = 13\), once you have applied the natural logarithm to both sides and used the power rule, you are left with a simpler equation: \(x \cdot \ln(5) = \ln(13)\).To solve for \(x\):

• Isolate \(x\) by dividing both sides by \(\ln(5)\): \(x = \frac{\ln(13)}{\ln(5)}\).
• Calculate the natural logarithms: \(\ln(13) \approx 2.5649\) and \(\ln(5) \approx 1.6094\).
• Perform the division: \(x \approx \frac{2.5649}{1.6094} \approx 1.5933\).

This step-by-step approach not only provides a solution but also demonstrates the logical flow of solving such equations. The key lies in breaking down the problem using logarithmic properties and then applying basic arithmetic to find your answer.