Problem 38 - 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

Exponential form: \( m^{101} = P \).

Step 1: Identify the Components

In the given logarithmic equation \( \log_{m} P = 101 \), identify the base of the logarithm, the argument, and the result. Here, \( m \) is the base, \( P \) is the argument, and 101 is the result.

Step 2: Recall Logarithmic Definition

Remember that a logarithmic equation \( \log_{b} a = c \) can be rewritten as an exponential equation \( b^{c} = a \). This is the definition of a logarithm: the power to which the base must be raised to produce the argument.

Step 3: Convert to Exponential Form

Using the definition from Step 2, convert the logarithmic equation \( \log_{m} P = 101 \) to its exponential form: \( m^{101} = P \). This shows the relationship between the base, the power (result), and the argument.

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve a logarithm with a variable inside. Understanding them requires knowing how to work with logarithms, which are the opposite of exponentials. Essentially, logarithmic equations allow us to tackle problems where the variable is an exponent.
When dealing with these equations, it’s crucial to identify key components like:

• Base: The number that is being raised to a power.
• Argument: The result or product of raising the base by the power—it’s what the argument evaluates to.
• Result: The value that the base’s power must equal to obtain the argument.

For example, in the equation \( \log_{m} P = 101 \), \( m \) is the base, \( P \) is the argument, and the result is 101. This equation tells us that the base \( m \), raised to the 101st power, equals \( P \). Recognizing these components is the first step in converting a logarithmic equation to an exponential form.

Exponential Form

Understanding exponential form is essential when it comes to rewriting logarithmic equations. Exponential form allows us to express mathematical relationships clearly, especially when implying large or complex operations.
An exponential equation like \( b^c = a \) expresses that the base \( b \), raised to the power of \( c \), equals \( a \). This is straightforward when you are familiar with exponentiation.
To convert a logarithmic equation like \( \log_{m} P = 101 \) into an exponential one, we use the expression derived from its definition: \( m^{101} = P \). This format displays the power relationship directly, giving a vivid picture of the equation's core components: base, exponent, and result.
A clear understanding of exponential form provides an important foundation, making it easier to transition between logarithmic and exponential representations and solve related problems.

Logarithm Properties

The properties of logarithms are instrumental in solving complex mathematical problems involving exponential and logarithmic expressions. They help simplify expressions and equations, making the problem-solving process easier. Here are some key properties:

• Product Property: \( \log_b (xy) = \log_b x + \log_b y \).
• Quotient Property: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
• Power Property: \( \log_b (x^y) = y \cdot \log_b x \).

These properties show how logarithms behave under different mathematical operations. For instance, the power property can bust complex powers into simpler multiplication.
Understanding these properties helps not only in converting and manipulating logarithmic and exponential forms but also in solving real-world problems where growth and decay processes are examined, like population growth and radioactive decay. Keeping these properties in mind will make working with logarithms less daunting and much more intuitive.