Problem 22 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Exponential Equation

The equation provided is \( \left(\frac{7}{5}\right)^{m} = n \). This is in the exponential form \( a^b = c \), where \( a = \frac{7}{5} \), \( b = m \), and \( c = n \).

Step 2: Understand the Relationship of Exponential and Logarithmic Functions

To convert from exponential form to logarithmic form, use the identity: If \( a^b = c \), then \( b = \log_a(c) \). This means \( a \) is the base, \( b \) is the exponent, and \( c \) is the result.

Step 3: Write the Equation in Logarithmic Form

Using the exponential-to-logarithmic relationship, write \( \left(\frac{7}{5}\right)^{m} = n \) in logarithmic form as \( m = \log_{\frac{7}{5}}(n) \).

Key Concepts

Exponential Equations

An exponential equation is a type of equation where a constant base is raised to a variable exponent. In our exercise, the equation \( \left(\frac{7}{5}\right)^{m} = n \) is exponential. Here, the base is \( \frac{7}{5} \), the exponent is \( m \), and the result is \( n \). These types of equations frequently appear in many areas, including growth and decay problems in physics, finance, and biology.
Understanding exponential equations is crucial because they often model real-world phenomena. Recognizing the base, the exponent, and the resultant value helps in solving and transforming these equations into other forms, like logarithmic equations. It's all about understanding how repeated multiplication (exponents) can be expressed in different mathematical terms.

Change of Base Formula

The change of base formula helps when we are converting logarithms from one base to another. Although this concept is not directly used in rewriting an exponential equation as a logarithm, it becomes vital if we need to calculate the logarithm using a calculator, which often only supports base 10 (common logarithm) or base \( e \) (natural logarithm).
The change of base formula is expressed as:

• \( \log_a(b) = \frac{\log_c(b)}{\log_c(a)} \)

Where \( c \) is the new base you are changing to, typically 10 or \( e \). This transforms the calculation into something manageable with standard tools, reinforcing that different bases can represent the same logarithmic value. Understanding this gives you versatility in solving logarithmic problems.

Logarithmic Form

Converting an exponential equation to logarithmic form involves a simple identity that equates the two forms. The identity states that if \( a^b = c \), then it can be rewritten as \( b = \log_a(c) \). For the exercise \( \left(\frac{7}{5}\right)^{m} = n \), this transforms to \( m = \log_{\frac{7}{5}}(n) \).
This conversion is pivotal because it allows easier manipulation and solution of equations, particularly when the variable is in the exponent. Logarithmic form expresses the exponent as the subject, denoting it as the output of the logarithmic function. It is an essential tool in both simplifying complex equations and solving for unknowns that would otherwise be tough to isolate in exponential form.

Algebraic Manipulation

Algebraic manipulation refers to the process of transforming and simplifying expressions to solve or rearrange equations. When dealing with exponential and logarithmic equations, algebraic manipulation is crucial in expressing one form in terms of another. It often involves applying different algebraic properties of exponents and logarithms.
For example, using the properties of logarithms, one can break down complex logarithmic expressions or simplify equations for easier computation. This might include using product, quotient, or power rules for logarithms, depending on what form is necessary for solving the equation.
This skill is invaluable for students as it enhances their ability to solve equations by rearranging terms, factoring, expanding, and simplifying. Such manipulation also equips them to handle more advanced and multifaceted mathematical problems efficiently.