Problem 13 - A Graphical Approach to College Algebra

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. $$0.8^{x}=4$$

Step-by-Step Solution

Answer

Exact form: \(x = \frac{\log(4)}{\log(0.8)}\); Approximation: \(x \approx -6.643\).

Step 1: Understand the Problem

We need to solve the exponential equation \(0.8^x = 4\). The goal is to express the solution in exact form and, if it's irrational, approximate it to the nearest thousandth.

Step 2: Take the Logarithm of Both Sides

Taking the logarithm allows us to solve for \(x\). Apply a logarithm (common or natural) to both sides of the equation: \(\log(0.8^x) = \log(4)\).

Step 3: Use the Power Rule of Logarithms

Apply the power rule \(\log(a^b) = b \cdot \log(a)\) to simplify the left side of the equation: \(x \cdot \log(0.8) = \log(4)\).

Step 4: Solve for x

Isolate \(x\) by dividing both sides by \(\log(0.8)\): \(x = \frac{\log(4)}{\log(0.8)}\).

Step 5: Exact Form of the Solution

The exact form of the solution is \(x = \frac{\log(4)}{\log(0.8)}\). This expression is in exact form using logarithms.

Step 6: Use a Calculator to Evaluate to the Nearest Thousandth

Approximate the value of \(x\) by evaluating the expression \(x = \frac{\log(4)}{\log(0.8)}\) using a calculator. This yields \(x \approx -6.643\).

Key Concepts

Logarithms

Logarithms are essential tools in mathematics for solving exponential equations like the one in this exercise, where the unknown, here represented by \( x \), appears as an exponent. In the equation \( 0.8^x = 4 \), a direct solution is not always obvious, hence we use logarithms.
Logarithms help by transforming exponential equations into linear ones, through the property that \( \log_b(a^c) = c \cdot \log_b(a) \). This conversion is useful because it allows us to solve for \( x \) using basic algebra.
You can use either common logarithms (base 10) or natural logarithms (base \( e \)), as they will both yield the same result when used consistently throughout the equation. However, in this example, we use the common logarithm so the calculation becomes \( \log(0.8^x) = \log(4) \), simplifying to \( x \cdot \log(0.8) = \log(4) \) thanks to the power rule.

• Logarithms are crucial for transforming complex exponential equations into manageable forms.
• They involve properties like the power rule, which aids in reducing the complexity of equations.

Irrational Numbers

Irrational numbers are numbers that cannot be exactly expressed as a simple fraction. They are real numbers that go on indefinitely without repeating. In the context of solving our mathematical problem, they often arise when we take logarithms.
When solving \( x = \frac{\log(4)}{\log(0.8)} \), the resulting value for \( x \) turns out to be an irrational number. We cannot write it as a precise fraction or a fixed decimal. This is why we use the concept of approximation, often expressed to a certain decimal place for precision, like in this exercise.
Approximating irrational numbers makes them easier to work with in practical applications, even though it's important to remember that they are not exact values.

• Irrational numbers arise when continuous numbers are represented without an exact fractional representation.
• Approximation allows us to deal with these numbers in a manageable way by rounding to decimal places.

Exact Form Solution

Expressing solutions in exact form is another key concept in solving exponential equations. Here, the exact form retains the components of the operation, showing the true mathematical relationship without rounding or estimating.
In our equation, this was achieved by keeping the solution as \( x = \frac{\log(4)}{\log(0.8)} \). This form is beneficial for symbolic manipulation and remains accurate no matter what precision is needed later.
However, exact form solutions might not always yield easy-to-handle numbers, especially when calculators are used, which is why approximation is also significant in the same exercise. It provides a tangible value that can be used in specific situations, maintaining usability and precision in calculations.

• Exact form solutions maintain the purity of the mathematical relationship and allow for further algebraic operations without loss of precision.
• They represent the true mathematical relationship, avoiding premature rounding.