Problem 15 - Algebra for College Students

Question

Write each logarithmic statement in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{10} 10,000=4 $$

Step-by-Step Solution

Answer

\(10^{4} = 10,000\).

Step 1: Understand the Logarithmic Statement

The logarithmic statement provided is \( \log_{10} 10,000 = 4 \). This statement tells us that 10,000 is 10 raised to the power of 4.

Step 2: Identify the Base, Exponent, and Result

In the logarithmic statement \( \log_{10} 10,000 = 4 \), the base is 10, the exponent is 4, and the result is 10,000. This means that 10 raised to the power of 4 equals 10,000.

Step 3: Write in Exponential Form

Now we convert the logarithmic form to exponential form using the relationship: if \( \log_{b} a = c \), then \( b^{c} = a \). Thus, \( 10^{4} = 10,000 \).

Key Concepts

Base of a Logarithm

In any logarithmic expression, the base is a fundamental component. It is the number that is raised to a power in order to get another number. For instance, consider the expression \( \log_{10} 10,000 = 4 \). Here, the base is 10. This means we're determining what power 10 needs to be raised to in order to receive the result of 10,000.

• The base number is found as the subscript in a logarithm: \( \log_{b} \ldots \)
• It represents the number being multiplied by itself a certain number of times.
• In our given example, the base 10 is being raised to the power determined by the exponent.

Understanding the base is crucial when working with logarithms as it sets the framework for determining the value needed for the result.

Exponent in Logarithms

The exponent in a logarithm indicates how many times the base is multiplied by itself to reach the given result. In our logarithm example, \( \log_{10} 10,000 = 4 \), the exponent is 4. This number tells us that the base, 10, must be multiplied by itself four times to achieve the result of 10,000.

• The exponent is always found after the equals sign in a logarithmic expression: \( \log_{b} a = c \)
• It expresses the power to which the base is raised.
• This helps us determine the relationship between the base and the resultant number.

By understanding the exponent, we can easily figure out the multiplication process needed to use the base to reach the intended result.

Convert Logarithms to Exponentials

Converting a logarithmic form to an exponential form involves rearranging the components of the expression. The conversion tells us how the base raised to the power of the exponent results in the given number. Using the example \( \log_{10} 10,000 = 4 \), we convert this to exponential form as \( 10^{4} = 10,000 \).

• In exponential form, the base is raised to the exponent: \( b^{c} = a \)
• This conversion provides a clearer view of the multiplication process behind the result.
• It simplifies understanding the dynamics of the logarithmic statement's components.

By mastering the conversion, you'll gain confidence in switching seamlessly between logarithmic and exponential expressions, enhancing both clarity and comprehension in mathematical discussions.