Problem 34 - Algebra 2

Question

Write each equation in logarithmic form. \(2401^{\frac{1}{4}}=7\)

Step-by-Step Solution

Answer

The logarithmic form is \(\log_{2401} 7 = \frac{1}{4}\).

Step 1: Understand the Problem Statement

The given equation is in exponential form: \(2401^{\frac{1}{4}}=7\). This means "2401 raised to the power of \(\frac{1}{4}\) gives 7." The goal is to express this equation in logarithmic form.

Step 2: Identify the Components of the Exponential Equation

In an exponential equation of the form \(a^b = c\), \(a\) is the base, \(b\) is the exponent, and \(c\) is the result. For our equation \(2401^{\frac{1}{4}}=7\), the base \(a = 2401\), the exponent \(b = \frac{1}{4}\), and the result \(c = 7\).

Step 3: Convert to Logarithmic Form

The logarithmic form of an exponential equation \(a^b = c\) is \(\log_a{c} = b\). For \(2401^{\frac{1}{4}} = 7\), we want to find \(\log_{2401} 7\), which should equal the exponent \(\frac{1}{4}\).

Step 4: Write the Logarithmic Equation

Applying the conversion to logarithmic form from Step 3, the equation \(2401^{\frac{1}{4}}=7\) can be rewritten as \(\log_{2401} 7 = \frac{1}{4}\). This means "the power to which 2401 must be raised to get 7 is \(\frac{1}{4}\)."

Key Concepts

Exponential Equations

Exponential equations are equations where the unknown appears as the exponent of some constant. These equations are written in the form \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result or output. For instance, in the equation \(2401^{\frac{1}{4}} = 7\), \(2401\) is the base, \(\frac{1}{4}\) is the exponent, and \(7\) is the result.

Understanding exponential equations is crucial because they model many real-world situations, such as compound interest, radioactive decay, and population growth. They represent a rapid way of increasing quantities and are foundational in the study of other mathematical topics, such as logarithms.

Logarithms

Logarithms are the inverses of exponential functions. When you take the logarithm of a number, you are essentially finding the exponent that a base must be raised to in order to yield that number. This is akin to "undoing" an exponentiation.

For example, if we know that \(a^b = c\), the logarithmic form is \(\log_a{c} = b\). In our original problem, converting an exponential equation like \(2401^{\frac{1}{4}} = 7\) into its logarithmic form results in \(\log_{2401} 7 = \frac{1}{4}\). This tells us that raising 2401 to the power of \(\frac{1}{4}\) gives us 7.

Logarithms are extremely useful for simplifying multiplication and division, understanding scales like the Richter scale for earthquakes, and solving for unknowns in exponential equations.

Base and Exponent Conversion

Converting between exponential and logarithmic forms involves understanding the relationship between the base, exponent, and result. This conversion process changes an equation from its form \(a^b = c\) to \(\log_a{c} = b\).

To perform this conversion, follow these steps:

• Identify the base \(a\), the exponent \(b\), and the result \(c\).
• Use the fact that \(a^b = c\) translates into \(\log_a{c} = b\).

Our particular example \(2401^{\frac{1}{4}}=7\) converts smoothly into logarithmic form as \(\log_{2401}7 = \frac{1}{4}\). Such conversions are vital for solving mathematical problems that deal with exponential growth or decay, as they help to define and manipulate these relationships more clearly.