Problem 57 - 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) \(9^{x}=1\) (b) \(10^{x}=\sqrt{10}\) (c) \(4^{x}=\sqrt[3]{4}\)

Step-by-Step Solution

Answer

(a) x=0, (b) x=0.5, (c) x≈0.33

Step 1: Solve for x in 9^x=1

To solve the equation \(9^x = 1\), recognize that any number raised to the power of 0 equals 1, i.e., \(a^0 = 1\) for any non-zero \(a\). Therefore, \(x = 0\).

Step 2: Solve for x in 10^x=sqrt(10)

Write the square root as an exponent: \(\sqrt{10} = 10^{1/2}\). Thus, the equation becomes \(10^x = 10^{1/2}\). Since the bases are the same, we equate the exponents: \(x = \frac{1}{2}\).

Step 3: Solve for x in 4^x=cbrt(4)

Write the cube root as an exponent: \(\sqrt[3]{4} = 4^{1/3}\). The equation becomes \(4^x = 4^{1/3}\). By matching the exponents (since the bases are equal), we find \(x = \frac{1}{3}\).

Key Concepts

Change of Base Formula

Understanding the change of base formula is crucial when dealing with exponential equations that do not have the same base. The formula helps you rewrite logarithms in terms of another base, facilitating easier calculations especially when using calculators for non-integer solutions. The change of base formula states that for any positive numbers \(a\), \(b\), and \(x\), with both \(a\) and \(b\) not equal to 1, the logarithm base \(a\) of \(x\) can be rewritten as:

• \(\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\)

In many cases, you'll change the base to 10 (common logarithm) or \(e\) (natural logarithm), because these are readily available buttons on most scientific calculators.
For example, if you're faced with an equation like \(2^x = 7\) and you need to solve for \(x\), you might change the base of the logarithms to make the problem easier to manage. Thus, \(x = \frac{\log_{10}(7)}{\log_{10}(2)}\), allowing for a straightforward calculation.

Exponent Rules

Exponent rules are foundational principles that simplify the handling of expressions involving exponents. These rules apply to all exponential functions and are crucial when solving equations or simplifying expressions.
Here are some basic exponent rules:

• Product of powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of powers: \(\frac{a^m}{a^n} = a^{m-n}\), provided \(a eq 0\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Power of a product: \((ab)^n = a^n \cdot b^n\)
• Zero exponent: \(a^0 = 1\), where \(a eq 0\)
• Negative exponent: \(a^{-n} = \frac{1}{a^n}\)

These rules allow us to manipulate and solve exponential expressions effectively. For instance, in the equation \(9^x = 1\), we use the rule that any number to the zero power is 1, leading us directly to the solution \(x = 0\). Understanding these rules ensures clarity and precision in solving exponential equations.

Solving for x

Solving exponential equations involves finding the value of the variable that makes the equation true. The method of solving varies based on the problem, often requiring you to simplify the equation to allow direct comparison of exponents.
To solve equations like those given:

• For \(9^x = 1\), recognize that any number raised to the zero power equals 1; hence, \(x = 0\).
• In \(10^x = \sqrt{10}\), express \(\sqrt{10}\) as \(10^{1/2}\). Because the bases (10) match, we set the exponents equal, giving \(x = \frac{1}{2}\).
• For \(4^x = \sqrt[3]{4}\), rewrite \(\sqrt[3]{4}\) as \(4^{1/3}\); equal bases on both sides allow equating exponents, so \(x = \frac{1}{3}\).

The key is often recognizing how to express one side of the equation in similar terms as the other to match the exponents easily. In more complex cases, looking up to change of base formula might become necessary.