Problem 12 - Algebra A Combined Function

Question

Write each as an exponential equation. $$ \log _{11} \sqrt[4]{11}=\frac{1}{4} $$

Step-by-Step Solution

Answer

The exponential equation is \( 11^{\frac{1}{4}} = \sqrt[4]{11} \).

Step 1: Understand the Given Logarithmic Equation

The given equation is \( \log_{11} \sqrt[4]{11} = \frac{1}{4} \). Here, the logarithm \( \log_{11} \sqrt[4]{11} \) is expressed as having a base of 11 and a result of \( \frac{1}{4} \). This signifies that 11 raised to the power of \( \frac{1}{4} \) equals \( \sqrt[4]{11} \).

Step 2: Convert the Logarithmic Equation into an Exponential Form

Using the definition of a logarithm, \( \log_b a = c \) implies that \( b^c = a \). Applying this to our equation, we rewrite it as an exponential equation: \( 11^{\frac{1}{4}} = \sqrt[4]{11} \).

Step 3: Verify the Conversion

The expression \( 11^{\frac{1}{4}} \) represents the fourth root of 11, which matches \( \sqrt[4]{11} \). Hence, the conversion from logarithmic to exponential form is accurate.

Key Concepts

Logarithmic Equations

Logarithmic equations involve expressions that use a logarithm, which is a powerful mathematical tool.
They can help us find unknown exponents and are especially useful in solving exponential equations.
In general, a logarithm tells us how many times we need to multiply a number (the base) to get another number.

• For example, in the equation \( \log_{11} \sqrt[4]{11} = \frac{1}{4} \), the base is 11.
• The expression \( \sqrt[4]{11} \) is the result of raising this base to a certain power.

The equality tells us that if we raise 11 to the power of \( \frac{1}{4} \), we will obtain \( \sqrt[4]{11} \). Understanding this helps in transitioning to other mathematical forms.

Exponential Form

The exponential form is a method of expressing a position of power or exponent in equations.
Moving from a logarithmic equation to an exponential form can clarify the transformation of values and reveal what the numbers represent.
Here is how the conversion happens:

• The basic definition is \( \log_b a = c \) means \( b^c = a \).
• Taking our problem \( \log_{11} \sqrt[4]{11} = \frac{1}{4} \), we convert it into the exponential form: \( 11^{\frac{1}{4}} = \sqrt[4]{11} \).

The exponential form is sometimes easier to work with because it lays out exactly what operations need to be performed on the base number.

Conversion of Equations

Converting between forms—such as from logarithmic to exponential—is a common task in algebra and helps uncover different properties of numbers involved.
This process aids in simplifying complex equations or solving them more effectively. Here is why this conversion is vital:

• It enables the direct application of powers and roots.
• Illustrates the relationship between the base and the exponent clearly.

In the context of our exercise, converting \( \log_{11} \sqrt[4]{11} = \frac{1}{4} \) to \( 11^{\frac{1}{4}} = \sqrt[4]{11} \) helps us see and verify the equivalence of the numbers involved. This skill is not just essential for solving homework problems but also for tackling real-world scenarios like growth models or data algorithms.

Logarithms

A logarithm is essentially the opposite of exponentiation. While exponentiation involves raising a base to a certain power, logarithms take the result of that process and find the power.
They are extremely useful for simplifying large numbers and can make complex calculations more manageable.

• For example, \( \log_{11} \sqrt[4]{11} = \frac{1}{4} \) reveals how the logarithm connects a base (11), a power (\( \frac{1}{4} \)), and their result (\( \sqrt[4]{11} \)).
• Understanding logarithms helps in solving exponential equations and functions by rearranging terms.

Learning how to manipulate and understand logarithms opens the door to deeper mathematical insight and computational efficiency, useful across science, engineering, and beyond.