Problem 29 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{5} \sqrt{5}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The exponential equation is \( 5^{\frac{1}{2}} = \sqrt{5} \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{5} \sqrt{5} = \frac{1}{2} \). In a logarithmic equation \( \log_{b}(x) = y \), \( b \) is the base, \( x \) is the argument, and \( y \) is the result of the logarithm. This means \( b^{y} = x \).

Step 2: Identify Components

In the equation \( \log_{5} \sqrt{5} = \frac{1}{2} \), the base \( b \) is 5, the argument \( x \) is \( \sqrt{5} \), and the result \( y \) is \( \frac{1}{2} \).

Step 3: Write the Exponential Form

According to the logarithmic identity \( b^{y} = x \), write the exponential form. Replace \( b \) with 5, \( y \) with \( \frac{1}{2} \), and \( x \) with \( \sqrt{5} \): \[ 5^{\frac{1}{2}} = \sqrt{5} \]

Key Concepts

Exponential Equations

Exponential equations are fundamental in mathematics. These equations involve variables in the exponent. For example, the equation \(5^{x} = 25\) has "5" as the base and "x" as the exponent. To solve exponential equations, you often use logarithms to express it in a simpler form. Breaking down an exponential equation involves comparing the bases on both sides or transforming it into a corresponding logarithmic equation. This can effectively help to solve for the variable. Understanding how exponential equations work will build a strong foundation for advanced topics like inverse functions and growth models.

Logarithmic Identities

Logarithmic identities are the basic rules that simplify complex logarithmic expressions. Logarithms serve as the inverse operation of exponentiation and follow similar properties for manipulation. Here are some essential identities to remember:

• \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\)
• \(\log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n)\)
• \(\log_{b}(m^n) = n \cdot \log_{b}(m)\)
• \(\log_{b}(b) = 1\)
• \(\log_{b}(1) = 0\)

These identities are useful in transforming logarithmic equations into a more readable form. They allow you to break down or combine log expressions for ease of solving. For instance, in converting the logarithm \(\log_{5} \sqrt{5} = \frac{1}{2}\) into an exponential form, the identity \(b^{y} = x\) simplifies understanding the relationship between the logarithm and its components.

Understanding Logarithms

Understanding logarithms begins by recognizing them as tools for solving exponential equations. Logarithms answer the question: "What power should the base be raised to, in order to obtain a number?" For instance, in \(\log_{5}(25) = 2\), 5 to the power of 2 yields 25. Thus, the logarithm is 2. They are useful for dealing with very large or small numbers, making them integral in fields like engineering and the natural sciences. Key foundational concepts involve grasping the base ("b" in \(\log_{b}(x)\)), the argument ("x"), and the result ("y"). Converting a logarithmic equation like \(\log_{5} \sqrt{5} = \frac{1}{2}\) into its exponential form \(5^{\frac{1}{2}} = \sqrt{5}\) depicts this transformation clearly. With this understanding, logarithms provide a bridge between different forms of mathematical expression, allowing for versatile problem-solving approaches.