Problem 38 - Elementary and Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{m} P=101 $$

Step-by-Step Solution

Answer

The equation \( \log_m P = 101 \) becomes \( m^{101} = P \) in exponential form.

Step 1: Understand the Logarithmic Equation Structure

The given equation is in logarithmic form: \( \log_m P = 101 \). Here, \( m \) is the base of the logarithm, \( P \) is the argument, and \( 101 \) is the result of the logarithm.

Step 2: Recall the Relationship with Exponents

The logarithmic equation \( \log_m P = 101 \) means that the base \( m \) raised to the power of the result (101) is equal to the argument \( P \). The general relationship is \( \log_b A = C \), which is equivalent to \( b^C = A \).

Step 3: Convert to Exponential Form

Using the relationship from Step 2, rewrite the logarithmic equation \( \log_m P = 101 \) in exponential form. It becomes \( m^{101} = P \), showing that \( m \) raised to the power 101 equals \( P \).

Key Concepts

Understanding Exponential Equations

Exponential equations are mathematical expressions where a variable appears in the form of an exponent. For example, in the equation \( m^{101} = P \), the variable "\( m \)" is raised to the power of 101. Exponential equations often appear when dealing with growth processes such as population growth or radioactive decay. These equations can seem complex because the unknown, which we are trying to find, resides in the exponent.

However, they are invaluable in linking two major fields of mathematics: logs and exponents. Here's how you can identify an exponential equation:

• The structure is typically in the form of \( a^x = b \), where \( a \) is the base, \( x \) is the exponent, and \( b \) is the result.
• Solving these equations often involves taking the logarithm of both sides to bring down the exponent into a more manageable form.

Understanding how to manipulate these equations provides a bridge to solving real-world problems involving rates of change and growth patterns.

Exploring Logarithm Properties

Logarithms are functions that allow us to determine the power to which a base number must be raised to obtain another number. Understanding the properties of logarithms is crucial for transforming and working with logarithmic and exponential equations. These properties serve to simplify and solve complex mathematical problems easily.

• Product property: \( \log_b(M \times N) = \log_b M + \log_b N \). This property states that the log of a product is the sum of the logs of the factors.
• Quotient property: \( \log_b(\frac{M}{N}) = \log_b M - \log_b N \). It indicates that the log of a quotient is the difference of the logs.
• Power property: \( \log_b(M^n) = n \cdot \log_b M \). Here, the exponent in a log can be brought out front as a multiplier.
• Change of base formula: \( \log_b A = \frac{\log_c A}{\log_c b} \). This helps to compute logs with a calculator for any given base by transforming it to a more convenient base like 10 or \( e \).

These properties make calculations and operations involving logarithms more efficient and manageable, especially when dealing with large numbers or complex equations.

Understanding the Base of Logarithms

The base of a logarithm is one of the fundamental components of logarithmic computations. It determines the factor by which we repeatedly multiply to get the argument of the logarithm. In the equation \( \log_m P = 101 \), "\( m \)" serves as the base.

Understanding how the base functions:

• The base can be any positive number except 1 because raising 1 to any exponent always returns 1, which limits its usefulness.
• Common bases include 10, referred to as the common logarithm, and \( e \) (approximately 2.718), known as the natural logarithm.
• The base influences the rate of growth in the corresponding exponential function. For example, if our base was larger than \( e \), it would suggest faster growth in exponential terms.

Choosing different bases allows us to model different types of exponential growth effectively. This adaptability makes it such a versatile and powerful mathematical function in real-world applications.