Problem 11 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{7} \sqrt{7}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The exponential equation is \( 7^{\frac{1}{2}} = \sqrt{7} \).

Step 1: Understand the Logarithmic Equation

The logarithmic equation given is \( \log_{7}(\sqrt{7}) = \frac{1}{2} \). This equation represents that \( 7 \) raised to some power equals \( \sqrt{7} \). The power is given as \( \frac{1}{2} \).

Step 2: Convert to Exponential Form

In general, the logarithmic form \( \log_{b}(x) = y \) can be written in exponential form as \( b^y = x \). Applying this to our equation: \( b = 7 \), \( y = \frac{1}{2} \), and \( x = \sqrt{7} \).

Step 3: Write Exponential Equation

Using the conversion from Step 2, \( 7^{\frac{1}{2}} = \sqrt{7} \). This is the exponential form of the given logarithmic equation.

Key Concepts

Exponential Form

Understanding exponential form is crucial when dealing with exponential and logarithmic equations. An equation in exponential form expresses a number as a base raised to an exponent. For example, when we say that \(a^b = c\), we mean that the base \(a\) raised to the power of \(b\) results in \(c\).

In our case, we have a familiar base of \(7\) which when raised to the power of \(\frac{1}{2}\) yields another mathematical entity - \(\sqrt{7}\). This means that instead of calculating something new, you can think of exponential equations as a way to express relationships using compact formulations.

Exponential forms provide a straightforward way to express very large or very small numbers, and understanding them is foundational for advanced mathematics. They serve as a bridge between purely numeric expressions and those requiring deeper mathematical manipulations.

Logarithmic Form

Logarithmic form allows us to express exponentiation in a different light. It essentially reverses the process, converting an exponential equation into one that emphasizes the exponent, often making it easier to solve for unknowns.

In our example, we converted the exponential equation \(7^{\frac{1}{2}} = \sqrt{7}\) into its logarithmic form \(\log_{7}(\sqrt{7}) = \frac{1}{2}\). Here, the base \(7\) becomes the base of the logarithm, \(\sqrt{7}\) is the result, and \(\frac{1}{2}\) is the exponent we are solving for.

The logarithm asks the question: "To what power do we need to raise the base to get this result?" Understanding this transformation is essential, especially in solving equations where the exponent is the unknown value.

• Logarithms simplify the process of dealing with exponential expressions.
• They are critical in various fields such as finance, science, and engineering.

Converting Equations

Converting equations from logarithmic form to exponential form, or vice versa, is a key skill in algebra. This translation helps in understanding and solving a wide range of mathematical problems.

The general formula for these conversions is quite straightforward:

• If you have a logarithmic form \(\log_{b}(x) = y\), it translates to the exponential form \(b^y = x\).
• Likewise, converting \(b^y = x\) to \(\log_{b}(x) = y\) brings out the power that transforms the base into the number x.

These conversions are particularly useful when you encounter logarithmic equations that need solving, as translating them back to exponential form often simplifies the process. Thus, gaining proficiency with conversions enhances your algebraic toolkit, allowing for greater flexibility and problem-solving prowess in mathematics.