Problem 26 - Precalculus

Question

In Exercises \(1-33,\) solve the equation analytically. $$ 7^{3+7 x}=3^{4-2 x} $$

Step-by-Step Solution

Answer

\( x = \frac{4 \ln(3) - 3 \ln(7)}{7 \ln(7) + 2 \ln(3)} \)

Step 1: Take the Logarithm of Both Sides

To solve the equation analytically, we need to take the logarithm of both sides. This is because logarithms allow us to bring down exponents. We will use the natural logarithm (ln) for simplicity: \[ \ln(7^{3+7x}) = \ln(3^{4-2x}) \]

Step 2: Apply the Power Rule of Logarithms

Use the logarithm power rule which states that \( \ln(a^b) = b \cdot \ln(a) \). Applying this rule to both sides, we get:\[ (3 + 7x) \cdot \ln(7) = (4 - 2x) \cdot \ln(3) \]

Step 3: Distribute the Logarithms

Distribute the logarithms to simplify the equation further:\[ 3 \ln(7) + 7x \ln(7) = 4 \ln(3) - 2x \ln(3) \]

Step 4: Group Terms Involving x

To solve for \(x\), group all terms containing \(x\) on one side of the equation:\[ 7x \ln(7) + 2x \ln(3) = 4 \ln(3) - 3 \ln(7) \]

Step 5: Factor Out x

Factor \(x\) out of the left side of the equation:\[ x (7 \ln(7) + 2 \ln(3)) = 4 \ln(3) - 3 \ln(7) \]

Step 6: Solve for x

Divide both sides by the expression \((7 \ln(7) + 2 \ln(3))\) to isolate \(x\):\[ x = \frac{4 \ln(3) - 3 \ln(7)}{7 \ln(7) + 2 \ln(3)} \]

Step 7: Simplify the Expression

Calculate the numerical values of the natural logarithms and simplify the expression, if given specific values or approximations are needed. For now, we leave it in the current form as a representation of the solution.

Key Concepts

Understanding Logarithms

Logarithms help us deal with exponential equations by converting multiplication into addition, which simplifies analysis tremendously. When you see an equation involving a term like \( a^b \), it can be tackled with logarithms due to their innate property of bringing down exponents. This transformation is crucial in solving equations analytically.

• In this problem, we're dealing with exponential terms on both sides of the equation: \( 7^{3+7x} = 3^{4-2x} \).
• The goal is to bring those exponents out of the exponential expression using logarithms.
• The concept works like this: if \( a^b = c \), taking the logarithm of both sides gives \( \log(a^b) = \log(c) \).

Breaking down the exponents helps isolate the variable and reduces complex multiplicative relationships into simpler additive ones.

The Role of the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm that uses the base \( e \) (approximately 2.718). We use the natural logarithm here to simplify our calculations. Natural logarithms are particularly beneficial because they frequently appear in calculus and complex mathematical operations.

• Given that we take \( \ln(7^{3+7x}) = \ln(3^{4-2x}) \), we can immediately apply simplifications.
• Natural logs have the same core properties as other logarithms, making them perfect for reducing the equation's complexity.

Employing \( \ln \) allows the calculation to proceed smoothly, avoiding unnecessary complications with other logarithm bases. It's a universally accepted method, especially in high-level mathematics.

How the Power Rule Simplifies Equations

The power rule in logarithms is a powerful tool used to simplify expressions. According to the power rule, \( \ln(a^b) = b \cdot \ln(a) \). It allows us to remove exponents and work with their coefficients directly, a crucial step in solving equations efficiently.

• For instance, applying to our problem: \( (3 + 7x) \cdot \ln(7) = (4 - 2x) \cdot \ln(3) \).
• Instead of dealing with the exponentiated terms directly, we expand and rearrange them into a more manageable linear equation.

By using the power rule, we shift the focus from complex exponential terms to straightforward linear relationships, helping in isolating and solving for variables like \( x \). It disentangles the relationships that initially seem tangled in exponential form.

Finding the Solution for x

The ultimate goal in our equation is to solve for the variable \( x \). After transforming both sides of the equation using logarithms and applying simplifications, we reach the algebraic expression with \( x \) terms grouped:

• We rewrite to group all terms involving \( x \): \( 7x \ln(7) + 2x \ln(3) = 4 \ln(3) - 3 \ln(7) \).
• With all terms involving \( x \) on one side, factoring \( x \) out gives: \( x(7 \ln(7) + 2 \ln(3)) = 4 \ln(3) - 3 \ln(7) \).
• Finally, solve for \( x \) by dividing both sides to yield: \( x = \frac{4 \ln(3) - 3 \ln(7)}{7 \ln(7) + 2 \ln(3)} \).

This formula provides the value of \( x \) expressed in terms of natural logarithms. It is left as an exact expression unless numerical approximation is required. Understanding this process showcases the power of analytical methods in dealing with exponential equations.