Problem 26 - Intermediate Algebra

Question

Write each logarithmic equation as an exponential equation. See Example 1. Do not solve. $$ \log _{10} 100=2 $$

Step-by-Step Solution

Answer

The exponential equation is \( 10^2 = 100 \).

Step 1: Understanding Logarithmic Form

The given equation is \( \log_{10} 100 = 2 \). In a logarithmic equation of the form \( \log_b a = c \), \( b \) is the base of the logarithm, \( a \) is the argument, and \( c \) is the value that the log equals. This indicates that \( b \) raised to the power of \( c \) equals \( a \).

Step 2: Rewriting in Exponential Form

To convert the logarithmic equation into an exponential equation, apply the rule: \( b^c = a \). Here, replace \( b \) with 10, \( c \) with 2, and \( a \) with 100.

Step 3: Finalize the Exponential Equation

Now rewrite the expression: \( 10^2 = 100 \). This equation represents the original logarithmic equation in its exponential form.

Key Concepts

Logarithmic Form

Logarithmic form is a way to express relationships between numbers using logs. In a logarithmic equation like \( \log_b a = c \), "log" tells us how many times the base \( b \) must be multiplied by itself to reach \( a \). Essentially, it answers the question: "\( b \) raised to what power equals \( a \)?"
This notation is widely used in various fields, such as mathematics, computer science, and engineering. You might encounter it in calculations involving complex growth rates, such as exponential growth, certain financial equations, or in algorithms for computing.
When reading a logarithmic form, focus on:

• The base \( b \), which is a consistent multiplier.
• The argument \( a \), the number you get when you multiply \( b \) by itself \( c \) times.
• The result \( c \), or the logarithm, showing the power to which you raise the base to get \( a \).

Understanding these parts will help you easily convert between forms.

Exponential Form

Exponential form is another way to show the relationship expressed by a logarithm. It takes the format \( b^c = a \), stating explicitly that "\( b \) raised to the power of \( c \) equals \( a \)."
This representation is intuitive for showing how numbers grow and are instrumental in understanding exponential growth, decay, and compounding phenomenon.
Exponential equations are found everywhere, from calculating compound interest in finance to modeling population growth in biology. They provide clarity by directly showing how a base number grows exponentially.
When dealing with exponential form:

• The base \( b \) is what gets repeatedly multiplied.
• The exponent \( c \) tells us how many times to multiply the base.
• The result \( a \) is what you get when multiplying \( b \), \( c \) times.

Once you grasp this concept, interpreting growth patterns or even solving equations becomes a breeze.

Base of Logarithms

The base of a logarithm is pivotal in both logarithmic and exponential expressions. In \( \log_b a = c \), \( b \) is the base, symbolizing the consistent multiplier or factor that when raised to power \( c \), results in \( a \).
Understanding the base helps reveal the relationship between numbers:

• A familiar base is 10, known as the common logarithm, which is often assumed if no base is written.
• Natural logarithms use \( e \) (approximately 2.718), and these are central in continuous growth calculations, such as those seen in calculus and natural sciences.
• Other bases can be chosen depending on contexts, like 2 in computer science for binary log calculations.

Knowing which base to use is crucial as it affects the resulting calculations and interpretations of logarithmic and exponential forms.

Converting Logarithms to Exponentials

Converting logarithmic equations to exponential form is a fundamental skill in mathematics, streamlining complex expressions. The key is to recognize that both notations express the same concept.
To convert from \( \log_b a = c \) to exponential form, use this approach:

• Identify the components: the base \( b \), the argument \( a \), and the result \( c \).
• Rewrite by raising the base \( b \) to the power of \( c \): \( b^c = a \).
• Verify with an example: For \( \log_{10} 100 = 2 \), rewriting gives \( 10^2 = 100 \), easily understandable as the original values satisfy the mathematical rule.

This conversion helps in solving equations, simplifying expressions, and enhancing understanding of exponential growth and decay. Master this, and you'll find it much easier to work with more complex mathematical problems.