Problem 58 - 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) \(2^{x}=\sqrt{8}\) (b) \(7^{x}=1\) (c) \(e^{x}=\sqrt[y]{e}\)

Step-by-Step Solution

Answer

(a) \(x = \frac{3}{2}\), (b) \(x = 0\), (c) \(x = \frac{1}{y}\).

Step 1: Solve part (a): Express with common bases

First, express both sides of the equation \(2^x = \sqrt{8}\) with the base of 2. Notice that \(\sqrt{8} = 8^{1/2} = (2^3)^{1/2} = 2^{3/2}\). The equation can now be written as \(2^x = 2^{3/2}\).

Step 2: Equate the exponents for part (a)

Since the bases are the same, we can set the exponents equal to each other: \(x = \frac{3}{2}\).

Step 3: Solve part (b): Express in exponential form

For the equation \(7^x = 1\), realize that any number to the power of 0 is 1. Therefore, \(7^0 = 1\).

Step 4: Equate the exponent for part (b)

Since \(7^x = 7^0\), we equate the exponents to find \(x = 0\).

Step 5: Solve part (c): Express with common bases

For the equation \(e^x = \sqrt[y]{e}\), realize that \(\sqrt[y]{e} = e^{1/y}\). Therefore, the equation becomes \(e^x = e^{1/y}\).

Step 6: Equate the exponents for part (c)

With the same base, equate the exponents: \(x = \frac{1}{y}\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which variables appear as exponents. These are used to model a wide range of real-world phenomena, such as population growth, radioactive decay, and financial investments.
Solving exponential equations often involves expressing the equation such that both sides have the same base. Once the bases are identical, the problem reduces to solving for the exponents. For example, in the equation \(2^x = \sqrt{8}\), we first express \(\sqrt{8}\) as \(2^{3/2}\). Therefore, the equation simplifies to \(2^x = 2^{3/2}\), allowing us to equate the exponents (i.e., \(x = 3/2\)).

• Understanding the properties of exponents can greatly simplify solving these equations.
• Matching bases can transform complicated equations into simple algebraic ones.

Change of Base Formula

The change of base formula is a useful tool in algebra, particularly when working with logarithms in exponential equations. It allows you to express logarithms in terms of any chosen base, which is especially handy when the bases aren't easily compatible.
For logarithms, the change of base formula is expressed as:\[\log_b{a} = \frac{\log_c{a}}{\log_c{b}}\]This formula is pivotal when using calculators to approximate solutions since most calculators only have logarithm keys for base 10 (log) or base \(e\) (ln).
For example, if solving \(7^x = 1\), one could use:

• The fact that \(7^0 = 1\), recognizing immediately that the solution is \(x=0\).
• The change of base formula helps in more complex scenarios beyond this example.

Solving Equations

Solving equations, particularly exponential ones, is a crucial skill in mathematics as it involves strategically manipulating the equation to isolate the variable. This often entails performing operations that simplify the equation step-by-step.
Consider the equation \(e^x = \sqrt[y]{e}\). By expressing \(\sqrt[y]{e}\) as \(e^{1/y}\), you can see that both sides of the equation share the base \(e\), allowing you to set the exponents equal to each other: \(x = 1/y\).
Key techniques include:

• Manipulating the expression to have the same base on each side.
• Replacing radical expressions with fractional exponents.
• Equating exponents to solve for the unknown variable.

Mastery of these techniques enables one to handle a variety of algebraic problems with exponential components.