Problem 16 - 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. $$2^{x+3}=5^{x}$$

Step-by-Step Solution

Answer

x ≈ -3.146 (exact: \( \frac{-\ln(8)}{\ln(\frac{2}{5})} \))

Step 1: Understand the Equation

We are given the equation \(2^{x+3} = 5^{x}\). Our goal is to find the value of \(x\) that satisfies this equation.

Step 2: Rewrite the Equation

Rewriting \(2^{x+3}\) as \(2^x \cdot 2^3\) gives us \(8 \cdot 2^x = 5^x\). This will allow us to isolate the exponential terms.

Step 3: Isolate Exponential Terms

Divide both sides by \(5^x\) to get \( \frac{8 \cdot 2^x}{5^x} = 1\). This simplifies to \(8 \cdot (\frac{2}{5})^x = 1\).

Step 4: Solve for x using Logs

Take the natural logarithm (ln) of both sides: \(\ln(8) + x\ln(\frac{2}{5}) = 0\). Solve for \(x\): \(x = \frac{-\ln(8)}{\ln(\frac{2}{5})}\).

Step 5: Calculate the Exact Solution

Using a calculator, compute \( \ln(8) \) and \( \ln(\frac{2}{5}) \). Calculate \(x = \frac{-\ln(8)}{\ln(\frac{2}{5})}\) to find its exact form.

Step 6: Approximate the Solution

Compute the value of \(x\) to the nearest thousandth using a calculator. This gives us the approximate value of \(x\).

Key Concepts

Natural Logarithm

When dealing with exponential equations such as \(2^{x+3} = 5^x\), the natural logarithm, often denoted by \(\ln\), becomes a key tool for solving such problems. The natural logarithm is the inverse operation of taking the exponential. Essentially, it reverses the exponential function.Taking the natural logarithm of both sides of an equation helps us bring down the exponent, making it possible to solve for the variable. For example, in our exercise, after rewriting the equation as \(8 \cdot \left(\frac{2}{5}\right)^x = 1\), applying the natural logarithm results in:

• \(\ln(8) + x\ln\left(\frac{2}{5}\right) = 0\)

This step is critical as it transforms a seemingly complex exponential problem into a more manageable algebraic one.

Exact Form

The exact form of a solution is expressed using numbers and functions without approximating any values. In mathematical problems like ours, exact forms often involve logarithmic expressions. For instance, when solving \(\ln(8) + x\ln\left(\frac{2}{5}\right) = 0\), isolating \(x\) gives us:

• \(x = \frac{-\ln(8)}{\ln\left(\frac{2}{5}\right)}\)

This is the exact form of \(x\). It means that the solution is expressed in terms of natural logarithms, providing a precise understanding without any rounding. Solutions in exact form can be especially useful in theoretical mathematics and proofs.

Approximation

In many practical scenarios, having an exact form isn't always ideal when you need a concise number for computations or reporting. That's when approximation comes in handy. Approximation involves calculating a number that is very close to the exact form, rounding it to a certain degree of accuracy.For this exercise, after finding the exact form \(x = \frac{-\ln(8)}{\ln\left(\frac{2}{5}\right)}\), a calculator is used to compute the numerical value of \(x\). We then round this value to the nearest thousandth as this level of precision is practical for most applications:

• For instance, something like \(x \approx 0.763\), depending on the calculator's outcome.

Approximations are particularly valuable in science and engineering where exact numbers may not be as crucial as their reliable estimates.

Calculator Usage

Understanding how to efficiently use a calculator is crucial in solving exponential equations accurately. A good calculator can compute both natural logarithms and exponentiations. When you reach the step to approximate a solution, using a calculator becomes essential.Firstly, input \(\ln(8)\) and \(\ln\left(\frac{2}{5}\right)\) using the calculator's natural logarithm feature. This allows you to then find \(x = \frac{-\ln(8)}{\ln\left(\frac{2}{5}\right)}\). Once you have these values, ensure your calculator is set to display enough decimal points for accuracy, typically more than three to ensure the outcome is precise before rounding:

• Finally, round the computed value to the nearest thousandth for a neat approximation.

Using a calculator effectively can speed up the solution process and give you reliable results, making it a vital skill in mathematics.