Problem 56 - College Algebra with Modeling and Visualization

Question

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. (a) \(2^{x}=9\) (b) \(10^{x}=\frac{1}{1000}\) (c) \(e^{x}=8\)

Step-by-Step Solution

Answer

(a) \(x \approx 3.17\); (b) \(x = -3\); (c) \(x \approx 2.08\).

Step 1: Recognize the given form and use logarithms to solve (a)

You have to solve the equation \(2^{x} = 9\). To isolate \(x\), take the logarithm of both sides. Using logarithms to solve for \(x\) gives:\[x = \log_{2}{9}\]Since calculators typically do not compute logarithms with arbitrary bases, you'll use the change of base formula:\[x = \frac{\log_{10}{9}}{\log_{10}{2}}\]

Step 2: Use calculator for change of base formula (a)

Now use a calculator to evaluate \( \log_{10}{9} \) and \( \log_{10}{2} \). \[x = \frac{0.954}{0.301} \approx 3.170\]Round to the nearest hundredth: \(x \approx 3.17\).

Step 3: Use properties of exponents for (b)

For \(10^{x} = \frac{1}{1000}\), recognize that \( \frac{1}{1000} \) can be rewritten as \(10^{-3}\). So:\[10^x = 10^{-3}\]Since the bases are the same and are equal for both expressions, set the exponents equal:\[x = -3\]

Step 4: Solve using natural logarithms for (c)

The equation is \(e^{x} = 8\). To solve for \(x\), take the natural logarithm (\(\ln\)) of both sides:\[\ln(e^{x}) = \ln(8)\]Using the property \(\ln(e^x) = x\), you have:\[x = \ln(8)\]

Step 5: Use calculator to approximate (c)

Calculate \(\ln(8)\):\[x \approx 2.0794\]Round the result to the nearest hundredth: \(x \approx 2.08\).

Key Concepts

Change of Base Formula

When working with logarithms, especially with bases not supported by common calculators, the change of base formula becomes a handy tool. This formula allows you to convert a logarithm to one with a base that your calculator can handle, typically base 10 or base e (natural logarithm).
To perform the change using the formula, \[\log_b{a} = \frac{\log_c{a}}{\log_c{b}}\] How it Works:

• The numerator, \(\log_c{a}\), represents the logarithm of the target number \(a\) using the new base \(c\).
• The denominator, \(\log_c{b}\), represents the logarithm of the original base \(b\) using the new base \(c\).

Utilizing this formula allows us to solve equations where the base of the logarithm is not directly accessible. For example, to solve \(2^x = 9\), we substitute using the change of base formula to convert the base to 10, yielding a more straightforward calculation on most calculators.
This simple yet powerful method enhances our ability to solve equations involving different bases effectively.

Exponential Functions

Exponential functions are mathematical expressions involving a constant base raised to a variable exponent. The general form of an exponential function is \(f(x) = a^x\), where \(a\) is the constant base and \(x\) is the exponent variable. These functions are crucial in many fields, including science, finance, and engineering.
Key Characteristics:

• The base \(a\) is a positive real number, and can't be 1.
• As \(x\) increases, the value of \(a^x\) swiftly increases if \(a > 1\), and decreases if \(0 < a < 1\).

Exponential functions exhibit unique properties. For instance, any non-zero number raised to the power of zero equals one: \(a^0 = 1\). Additionally, they possess a smooth and continuous curve with no bounds, reflecting rapid growth or decay.
In the provided exercise, when we handled equations like \(2^x = 9\) and \(10^x = \frac{1}{1000}\), we translated these exponentials to logarithmic equivalents for solving. Understanding the interaction of exponents and logarithms allows for solving these equations efficiently.

Natural Logarithms

Natural logarithms involve logarithms with the mathematical constant \(e\) (approximately 2.718) as the base. The notation for a natural logarithm is \(\ln(x)\), which represents the power you raise \(e\) to result in \(x\).
Natural logarithms simplify calculations and analysis in calculus and transcendental equations.
Why Use Natural Logarithms?

• The base \(e\) is chosen because of its unique mathematical properties, particularly in continuous growth models.
• It simplifies derivatives and integrals in calculus, making them practical in physical and biological sciences.

In our exercise, the equation \(e^x = 8\) required natural logarithms for solving. By taking the natural logarithm of both sides, we derived \(\ln(e^x) = \ln(8)\). Using the property that \(\ln(e^x) = x\), the equation simplifies to \(x = \ln(8)\), further computed to approximately 2.08 when rounded.
This example demonstrates the power of natural logarithms in reducing complexity and aiding in the resolution of exponential equations.