Problem 3 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(4^{5 / 2}=32\)

Step-by-Step Solution

Answer

Rewrite as \(\log_{4}(32) = \frac{5}{2}\).

Step 1: Understand the Given Equation

Identify the structure of the given exponential equation. The equation provided is an exponential equation: \(4^{5/2} = 32\), where \( b = 4 \), \( a = \frac{5}{2} \), and \( c = 32 \).

Step 2: Apply the Exponential to Logarithmic Conversion

Use the property \(b^{a} = c\) translates to \(\log_{b}(c) = a\). In this scenario, substitute \(b = 4\), \(a = \frac{5}{2}\), and \(c = 32\) into \(\log_{b}(c) = a\) to get the logarithmic form: \(\log_{4}(32) = \frac{5}{2}\).

Step 3: Verification

Verify the conversion by checking if \(4^{5/2} = 32\) holds true using the logarithmic form \(\log_{4}(32) = \frac{5}{2}\). Calculate \(4^{5/2}\) to ensure it equals 32 and confirm that the conversion was correctly applied.

Key Concepts

Exponential Equations

An exponential equation is a mathematical expression where variables appear as exponents. In most basic terms, it lets us express repeated multiplication in a compact way. When you see an equation like \(4^{5/2} = 32\), it tells you that 4, our base, is raised to the power of \(\frac{5}{2}\), which is the exponent. Knowing how to solve exponential equations is crucial because they model many real-life phenomena, such as population growth, radioactive decay, and compounding interest.
Common forms of exponential equations are written as \(b^a = c\). Here, "b" is the base, "a" the exponent, and "c" the result of the expression. In our example, \(4^{5/2} = 32\), where \(b = 4\), \(a = \frac{5}{2}\), and \(c = 32\). In solving or rewriting these equations, recognizing these components is essential. It's your first step to transforming between an exponential equation and its corresponding logarithmic form.

Logarithm

Logarithms are the inverse operations of exponents. They help answer the question: to what power must a given base be raised to yield a certain number? The expression \(\log_b(c)\) asks, "to what power should the base \(b\) be raised to produce the number \(c\)?"
Logarithms are essential in various fields because they simplify multiplication and division into addition and subtraction. Understanding logarithms makes it easier to handle large quantities and measurements, especially when dealing with exponential growth or decay.
In our converted example, \(\log_4(32) = \frac{5}{2}\), this implies that raising the base 4 to the power of \(\frac{5}{2}\) gives you 32. This helps us see the connection between exponential equations and logarithms, providing an alternative way to view exponential functions.

Exponential to Logarithmic Conversion

Converting between exponential and logarithmic form is a valuable skill in mathematics. For any equations of the format \(b^a = c\), you can translate it into logarithmic form using the relationship \(\log_b(c) = a\). This represents the foundation of understanding how closely exponential expressions are tied to logarithms.
Consider the equation \(4^{5/2} = 32\). According to our conversion property, this translates to \(\log_4(32) = \frac{5}{2}\). Here’s how it works: You take the base (4), and then determine what power (\(\frac{5}{2}\)) it must be elevated to reach the number 32.
Having this conversion capability means you can switch between perspectives based on the kind of problem you're tackling, whether you need to solve for a variable in exponent form or analyze the problem using logs. It's like having two complementary tools for problem-solving.

Properties of Logarithms

Properties of logarithms are a set of rules that help simplify complex logarithmic expressions. They create easier pathways for calculations and transformations within log-based math tasks.
Key properties include:

• Product Property: \(\log_b(M \cdot N) = \log_b M + \log_b N\) — useful for turning multiplication into addition.
• Quotient Property: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\) — simplifies division into subtraction.
• Power Property: \(\log_b(M^k) = k \cdot \log_b M\) — lets you pull powers out of logarithms.

These properties are based on basic principles of exponents and extend their flexibility. They make complex calculations manageable, as well as turning heavy computations into more digestible steps. Understanding these aids in manipulating and solving both logarithmic and exponential equations efficiently.