Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each equation in logarithmic form. \(\left(\frac{7}{5}\right)^{m}=n\)

Step-by-Step Solution

Answer

\( \log_{\frac{7}{5}}(n) = m \) is the logarithmic form.

Step 1: Identify the Components

Firstly, identify the components of the given exponential equation. In the given equation \( \left(\frac{7}{5}\right)^{m} = n \), the base is \( \frac{7}{5} \), the exponent is \( m \), and the result is \( n \).

Step 2: Use the Definition of a Logarithm

Recall that the definition of logarithms states that if \( a^b = c \), then it can be rewritten in logarithmic form as \( \log_a(c) = b \).

Step 3: Apply the Definition to the Equation

Using the definition from Step 2, rewrite the given equation \( \left(\frac{7}{5}\right)^{m}=n \) in logarithmic form. Here, \( a = \frac{7}{5} \), \( b = m \), and \( c = n \). Therefore, the logarithmic form is \( \log_{\frac{7}{5}}(n) = m \).

Key Concepts

Exponential Equation

An exponential equation is a mathematical expression where a constant base is raised to a variable exponent, resulting in a value.
An example of an exponential equation can be seen in the form \(a^b = c\), where \(a\) represents the base, \(b\) is the exponent, and \(c\) is the result.

This concept is crucial as it helps in modeling numerous real-life situations, like population growth, radioactive decay, and interest calculations.
If you have an equation like \(\left(\frac{7}{5}\right)^{m} = n\), then \(\frac{7}{5}\) is the base, \(m\) is the exponent, and \(n\) is the overall value you get.

Working with exponential equations often involves converting them into logarithmic form. This makes it easier to solve for variables, especially when they appear in the exponent position.

Definition of Logarithms

Logarithms are the inverse operation to exponentiation. They help you find the exponent that a given base must be raised to in order to obtain a particular result.
For instance, if \(a^b = c\), this relationship can be rewritten using a logarithm as \(\log_a(c) = b\).

The definition of logarithms is vital in transforming complex equations, especially when dealing with unknown exponents.
In the provided exercise, this concept allows you to express the exponential equation \(\left(\frac{7}{5}\right)^{m} = n\) in an alternative way: \(\log_{\frac{7}{5}}(n) = m\).
This transformation into logarithmic form helps in solving for \(m\) if \(n\) is known, simplifying the equation into a more manageable format.

Base of Logarithm

The base of a logarithm is the number that is raised to a power. It forms the foundation of the logarithmic equation.
When you say \(\log_a(c) = b\), the \(a\) in this context is the base. It corresponds to the same number in the exponential form \(a^b = c\).

In logarithmic equations, the base determines the "building block" for the power needed to reach the result \(c\).
For the example \(\log_{\frac{7}{5}}(n) = m\), the base is \(\frac{7}{5}\).
This means you need to raise \(\frac{7}{5}\) to the power \(m\) to reach the number \(n\).
Understanding the base is crucial for correctly transforming between exponential and logarithmic forms.