Problem 11 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ 4+3^{5 x}=8 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately 0.2519.

Step 1: Isolate the exponential term

Start with the given equation: \( 4 + 3^{5x} = 8 \). Subtract 4 from both sides to isolate the exponential term: \( 3^{5x} = 8 - 4 \). This simplifies to \( 3^{5x} = 4 \).

Step 2: Apply logarithms to both sides

Take the natural logarithm of both sides to solve for \(x\): \( \ln(3^{5x}) = \ln(4) \). Use the power rule of logarithms, which states \( \ln(a^b) = b\ln(a) \), to rewrite the left side as \( 5x \ln(3) \).

Step 3: Solve for x

Divide both sides by \(5\ln(3)\) to isolate \(x\): \( x = \frac{\ln(4)}{5\ln(3)} \).

Step 4: Calculate numerically

Calculate the value numerically to four decimal places using a calculator: \( x \approx \frac{\ln(4)}{5\ln(3)} \approx 0.2519 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are incredibly useful when working with exponential equations, like the one given in this exercise. They allow us to linearize the problem, making it more manageable. In essence, logarithms are the opposite of exponentiation. If you have an equation with a variable in an exponential place, logarithms, especially natural logarithms (which we'll talk about next), help us bring that variable down to a more accessible form.
For instance, given an expression like \( a = b^x \), applying a logarithm to both sides can transform it into a form where solving for \( x \) becomes straightforward. Using the property \( \ln(a^b) = b \cdot \ln(a) \) simplifies expressions by leveraging the power rule of logarithms.

• Exponential form: \( 3^{5x} = 4 \)
• Logarithmic form after applying \( \ln \): \( \ln(3^{5x}) = \ln(4) \)

Understanding logarithmic functions is key to unraveling exponential relationships and forms a foundation for further explorations in calculus and other advanced mathematics topics.

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are logarithms with a base of the irrational number \( e \) (approximately 2.718). They are "natural" in the sense that they are extremely useful in calculus, especially in differentiating and integrating exponential functions.
In the context of our exercise, natural logarithms help simplify exponential expressions to solve for unknown variables embedded within powers.

• Given equation: \( \ln(3^{5x}) = \ln(4) \)
• Apply power rule: \( \ln(3^{5x}) = 5x \cdot \ln(3) \)
• Solve for \( x \): \( x = \frac{\ln(4)}{5\cdot\ln(3)} \)

Using natural logarithms is particularly efficient in distinguishing the growth nature of exponential functions, making them an indispensable tool in both theoretical and practical problems.

Numerical Solution Methods

Numerical solution methods involve calculating exact or approximate values using tools like calculators or computational software. When algebraic manipulation reaches a point where further simplification is not feasible, numerical methods allow us to find a practical solution by performing and verifying arithmetic calculations.
In this exercise, after isolating \( x \), we use a calculator to determine the exact numerical value of the expression:

• Solve \( x = \frac{\ln(4)}{5\cdot\ln(3)} \)
• Use a scientific calculator: Calculate \( \ln(4) \) and \( \ln(3) \), then perform division
• Approximate result: \( x \approx 0.2519 \)

Numerical methods are vital in practice because they offer precise results that are often needed in scientific studies, engineering tasks, and economic forecasting. They bridge the gap between theoretical problem-solving and real-world application, providing concrete answers to complex mathematical problems.