Problem 12 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log (100)=2\)

Step-by-Step Solution

Answer

The exponential form of \( \log(100) = 2 \) is \( 10^2 = 100 \).

Step 1: Identify the Given Equation Form

Recognize that the given equation \( \log(100) = 2 \) is a logarithmic equation. In this context, the base of the logarithm is implied to be 10 since it is written without a base. Therefore, it is equivalent to \( \log_{10}(100) = 2 \).

Step 2: Formulate the Exponential Equation

Use the property from Theorem 6.2, given as \( b^a = c \) if and only if \( \log_b(c) = a \). Based on this property, convert the logarithmic form \( \log_{10}(100) = 2 \) to its exponential form. Here, \( b = 10 \), \( a = 2 \), and \( c = 100 \).

Step 3: Apply the Property

Using the property, translate \( \log_{10}(100) = 2 \) into an exponential equation. By the property \( b^a = c \), this becomes \( 10^2 = 100 \). This shows the equivalent exponential form of the original logarithmic equation.

Key Concepts

Exponential Equations

Exponential equations are an important class of mathematical equations where the variable appears in the exponent. These equations take the general form \[ b^x = c \]where \( b \) is the base, \( x \) is the exponent or power, and \( c \) is the result. In solving exponential equations, one seeks to determine the value of \( x \) that makes the equation true.

• One typical method involves rewriting the equation in logarithmic form, which can simplify finding unknowns.
• Understanding the basics of exponential equations is crucial as they appear in various scientific and financial contexts, such as calculating compound interest or modeling growth processes.

In the provided exercise, converting \( \log_{10}(100) = 2 \) to its exponential form produced \( 10^2 = 100 \). Here, the conversion helps emphasize that the exponent 2 indeed results in multiplying the base 10 by itself to yield 100.

Properties of Logarithms

Logarithms provide powerful tools to deal with exponential forms in mathematics. These tools include unique properties that help simplify and solve equations. Among the most common properties are:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).
• Power Rule: \( \log_b(M^k) = k \cdot \log_b(M) \).

These properties streamline computations in both pure mathematics and applied sciences. For example, they make multiplying numbers or raising numbers to large powers more manageable. In the exercise, the ability to convert between forms using the theorem property takes advantage of the logarithmic relationship's straightforward comprehension, particularly when the base is known or can be easily inferred.

Conversion Between Forms

Converting between exponential and logarithmic forms is a fundamental mathematical skill that aids in understanding relationships and solving equations. This conversion is based on the property characterized by the equivalence:\[ b^a = c \quad \text{if and only if} \quad \log_b(c) = a \]This means any exponential equation can be expressed as a logarithmic equation and vice versa. Understanding this conversion is pivotal for students as it:

• Enables easier manipulation of equations when solving for unknowns, particularly when dealing with very large or very small numbers.
• Provides deeper insight into the behavior of growth patterns, decay, or time-based changes observed in fields like biology, finance, and physics.

In the exercise, the logarithmic equation \( \log_{10}(100) = 2 \) illustrates how an implicit base can be assumed and used to convert back and forth without difficulty, highlighting the versatility and necessity of this conversion process.