Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$5^{x}=13$$

Step-by-Step Solution

Answer

Exact form: \( x \approx \frac{\ln(13)}{\ln(5)} \); Approximation: \( x \approx 1.363 \).

Step 1: Apply Logarithms

Start by taking the logarithm of both sides with any log base (commonly base 10 or base e). Here, we'll use natural logarithms for simplicity: \[ \ln(5^x) = \ln(13) \].

Step 2: Use the Power Rule of Logarithms

Apply the power rule for logarithms which states \( \ln(a^b) = b\ln(a) \):\[ x \ln(5) = \ln(13) \].

Step 3: Solve for x

Isolate \( x \) by dividing both sides by \( \ln(5) \):\[ x = \frac{\ln(13)}{\ln(5)} \].

Step 4: Calculate the Exact Value

Using the expression we derived, solve for the exact value of \( x \):\[ x \approx \frac{2.564949}{1.609438} \].

Step 5: Approximate the Solution

Use a calculator to find the numerical value, rounding it to the nearest thousandth:\[ x \approx 1.363 \].

Key Concepts

Logarithms

Logarithms are an essential concept in mathematics, especially when working with exponential equations. In simple terms, a logarithm asks the question, "To what power must a given base be raised, to obtain a specific number?" For instance, in the question involving the equation \(5^x = 13\), the logarithm helps us express the exponential relationship into a linear form, which is more straightforward to solve. By introducing logarithms, we convert an unknown exponent problem into a calculable format. The equation becomes manageable by taking the logarithm of both sides, either common logarithms (base 10) or natural logarithms (base \(e\)).

• Logarithm notation is written as \(\log_b(a)\), which answers "what is the exponent for base \(b\) to give \(a\)?"
• Base \(e\) logarithms, also called natural logarithms, are denoted as \(\ln(a)\).
• Using logarithms lets us handle various complex exponential forms easily.

Power Rule of Logarithms

The power rule of logarithms is a pivotal concept when solving exponential equations. It simplifies complex expressions involving exponents, turning them into multiplicative operations. This rule states that \(\ln(a^b) = b \ln(a)\). This transformation is crucial as it "brings down" the exponent, allowing us to access the variable directly.When applied, the power rule changes the problem from \(\ln(5^x) = \ln(13)\) to \(x \ln(5) = \ln(13)\). In this revised form, it becomes straightforward to isolate the variable \(x\). This is particularly useful because it turns an exponential equation into a simple algebraic equation.

• The rule takes advantage of logarithm properties to simplify calculations.
• It ensures that complex exponentials can be managed with basic arithmetic.

Natural Logarithms

Natural logarithms, denoted as \(\ln\), are logarithms with base \(e\), where \(e\) is approximately equal to 2.71828. They are incredibly useful in higher mathematics and natural sciences due to their unique mathematical properties. In the context of solving exponential equations, natural logarithms simplify the process by eliminating the need for conversion between different bases.Using natural logarithms in the equation \(5^x = 13\), translates directly to \(\ln(5^x) = \ln(13)\). The exponential expression is handled systematically using the base \(e\), making the subsequent algebraic manipulations more efficient.

• Natural logarithms are preferred for equations involving exponential growth and decay.
• They offer a consistent framework within calculus and algebra.

Exact Form Solutions

In mathematics, expressing solutions in exact form means expressing them as precisely as possible without numerical approximation. This is critical for maintaining the integrity of mathematical calculations, particularly in more complex problems. In the case of the equation \(5^x = 13\), the exact form is found by setting the expression \(x = \frac{\ln(13)}{\ln(5)}\). This enables us to provide an exact representation before any rounding or calculator usage. Having an expression in exact form allows later use for more precise computations or when exact integer values are necessary for further calculations. The exact form helps both in verifying accuracy and enhancing understanding of the solution process.

• Exact forms preserve mathematical precision up until the point of approximation.
• They are useful in theoretical discussions and advanced mathematical contexts where approximations may introduce errors.