Problem 27 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 5^{1 / 2}=\sqrt{5} $$

Step-by-Step Solution

Answer

\(\log_{5}(\sqrt{5}) = \frac{1}{2}\)

Step 1: Understand the Exponential Equation

The given equation is \(5^{1/2} = \sqrt{5}\). Here, the base is 5, the exponent is \(\frac{1}{2}\), and the result is \(\sqrt{5}\).

Step 2: Recall the Definition of Logarithms

A logarithm \(\log_b(x) = y\) is an equation that means \(b^y = x\). Thus, the logarithm defines the power \(y\) to which the base \(b\) must be raised to get \(x\).

Step 3: Apply the Concept to the Given Exponential Equation

You need to express the exponential equation \(5^{1/2} = \sqrt{5}\) in logarithmic form. Here, the base \(b\) is 5, the resultant \(x\) is \(\sqrt{5}\), and the exponent \(y\) is \(\frac{1}{2}\).

Step 4: Write the Logarithmic Equation

Based on the definition, write \(\log_{5}(\sqrt{5}) = \frac{1}{2}\), which shows the power to which the base 5 should be raised to get \(\sqrt{5}\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions in which a constant base is raised to a variable exponent. They are written in the form \( b^y = x \), where \( b \) is the base, \( y \) is the exponent, and \( x \) is the result of the base raised to the power of the exponent. In our exercise, the equation is \(5^{1/2} = \sqrt{5}\). Here, 5 is the base, and \(\frac{1}{2}\) is the exponent, which results in \(\sqrt{5}\). Understanding exponential equations is crucial because they regularly appear in scientific contexts, such as population growth and compound interest.
\( \) To solve or manipulate exponential equations, it helps to understand how they're structured. The exponent indicates how many times the base is being multiplied by itself. In our example, however, the exponent \(\frac{1}{2}\) stands for a square root, which is the same as raising the base to the power of 0.5, thus rendering the result as \(\sqrt{5}\).

Logarithm Definition

A logarithm is a way of expressing an exponent in another form and is denoted as \( \log_b(x) = y \). This expression means "the power \(y\) to which the base \(b\) must be raised to produce \(x\)." In simpler terms, it's the reverse operation of exponentiation.
\( \) Think of logarithms as asking the question: "To what power must I raise this base to get the desired number?" For instance, if you have \(b^y = x\), then \(\log_b(x) = y\). This relation offers a convenient way to solve exponential equations by converting them into logarithmic form, making it easier to handle and understand the relationship between the elements.

• Base \(b\) in both exponential and logarithmic form is the same.
• Exponent \(y\) in exponential becomes the result in logarithmic form.
• Result \(x\) in exponential form becomes the input in a logarithm.

Conversion Between Forms

Converting between logarithmic and exponential forms is a valuable skill in mathematics that simplifies the solution of equations. To convert an exponential equation like \(5^{1/2} = \sqrt{5}\) to its logarithmic form, use the formula \( \log_b(x) = y \).
Here, the parts of the conversion are:

• Base \(b\) is 5.
• Result \(x\) is \(\sqrt{5}\).
• Exponent \(y\) is \(\frac{1}{2}\).

So, the equation \(\log_5(\sqrt{5}) = \frac{1}{2}\) is the logarithmic form of \(5^{1/2} = \sqrt{5}\).
Understanding these conversions helps in various applications like solving equations, simplifying complex expressions, and more. It makes it possible to shift between forms, allowing for flexible manipulation of mathematical expressions. Always ensure the base remains consistent across both forms, as they represent the same mathematical entity. This consistency lets you transition smoothly from the idea of repeated multiplication in exponential form to an abstract representation of exponentiation in logarithmic form.