Problem 9 - 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. \(\log _{25}(5)=\frac{1}{2}\)

Step-by-Step Solution

Answer

The exponential form is \( 25^{\frac{1}{2}} = 5 \).

Step 1: Understanding the Given Equation

We are given the equation \( \log_{25}(5) = \frac{1}{2} \). This is a logarithmic equation which we need to rewrite in exponential form.

Step 2: Identifying Base, Exponent, and Result

According to the logarithmic equation \( \log_{b}(c) = a \), where the base \( b \) is 25, the argument \( c \) is 5, and the result \( a \) is \( \frac{1}{2} \). We need these values to rewrite the equation in exponential form.

Step 3: Rewriting as an Exponential Equation

Using the property \( b^{a} = c \), rewrite the logarithmic equation \( \log_{25}(5) = \frac{1}{2} \) as an exponential equation: \( 25^{\frac{1}{2}} = 5 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve a logarithm with a variable either inside the log function or equated to a log expression. These equations are pivotal in unraveling complex exponential relationships by expressing them in a more manageable linear form.
Logarithms are the inverses of exponents. For a given number, a logarithm asks the question: "To what exponent must the base be raised, to yield the argument of the log function?" For instance, the logarithmic equation given in the exercise, \( \log_{25}(5) = \frac{1}{2} \), is asking, "To what power must 25 be raised to yield 5?"

• Base (b): The number that is raised to a power. In our example, 25 is the base.
• Argument (c): The result we get when the base is raised to the power. Here, it is 5.
• Logarithmic Value (a): The power or exponent that the base is raised to, which is \( \frac{1}{2} \).

Logarithmic equations can be rewritten to their exponential forms to provide a clearer view of the underlying exponential relationship they represent.

Exponential Equations

Exponential equations are such that a variable appears in the exponent. These equations describe exponential growth or decay scenarios, encompassing a broad array of real-world applications such as population growth, radioactive decay, interest calculations, and more.

To solve exponential equations, particularly those involving logarithms, you convert them into logarithmic equations to facilitate easier manipulation and solution.

• Exponential Form: The expression \( b^{a}=c \) where \( b \) is the base, \( a \) is the exponent, and \( c \) is the result.
• Rewriting: As seen in the original step-by-step solution, by rewriting the logarithmic equation \( \log_{25}(5)=\frac{1}{2} \) into the exponential form \( 25^{\frac{1}{2}}=5 \), we can verify and understand the underlying relationship expressed through the base-exponent-result expression.

By leveraging this rewriting process, we can often discover relationships or solve problems pertaining to exponential scenarios with greater ease.

Properties of Logarithms

The properties of logarithms are crucial mathematical tools that simplify the handling of logarithmic expressions and equations. These properties allow us to scale, transform, and solve logarithmic equations effectively, which is essential when dealing with exponential and other related problems.
Some fundamental properties include:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^k) = k\log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \), useful for computing logs in calculators which typically have base 10 or e.

Use these properties to streamline calculations involving logarithms, and transform logarithmic expressions into equivalent forms that are easier to solve or interpret. By applying these properties adeptly, you can deftly navigate through complex problems requiring the use of logarithms.