Problem 16 - Intermediate Algebra

Question

For Problems \(11-20\), write each of the following in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{10} 100,000=5 $$

Step-by-Step Solution

Answer

The exponential form is \(10^5 = 100,000\).

Step 1: Identify the Base, Exponent, and the Result

In the logarithmic equation \( \log_{10} 100,000 = 5 \), the base is 10, the exponent is 5, and the result is 100,000. These are the components that will be used to write the equation in exponential form.

Step 2: Convert to Exponential Form

Convert the logarithmic form \( \log_{10} 100,000 = 5 \) into its corresponding exponential form. In exponential form, this will be written as \( 10^5 = 100,000 \). This shows that raising the base (10) to the exponent (5) results in 100,000.

Step 3: Verify the Exponential Equation

Calculate \( 10^5 \) to ensure it equals \( 100,000 \). Performing the calculation: \( 10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000 \). This confirms that the conversion from logarithmic to exponential form is correct.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical statements that relate the concepts of logarithms and exponents. A logarithm, usually written as \( \log_{b}(x) \), essentially asks the question: "To what power should we raise the base \( b \) to get \( x \)?" In these equations, the base \( b \) is always greater than zero and typically greater than one. When working with logarithmic equations, the key is to identify the base, the result, and the exponent effectively. For example, in the equation \( \log_{10} 100,000 = 5 \),

• The base is 10.
• The result is 100,000, or what the base should become when raised to the exponent.
• The exponent is 5, indicating the power to which the base must be raised to reach 100,000.

Recognizing these elements is crucial for converting the equation into exponential form, which can simplify the equation's interpretation and solution.

Base and Exponent

The base and exponent are central components of both logarithms and powers. Understanding these terms can make dealing with equations easier. The base in a logarithm is the number that is repeatedly multiplied. For example, in \( \log_{10} 100,000 \), the base is 10. It serves as the foundation of the logarithmic calculation.

The exponent represents how many times the base is used as a factor. In the example of \( \log_{10} 100,000 = 5 \),

• The exponent is 5. This means that when 10 is multiplied by itself 5 times, it equals 100,000.

Choosing the correct base and understanding the exponent is essential for accurately converting between forms and solving problems using these concepts confidently.

Conversion from Logarithmic to Exponential

Converting a logarithmic equation to its exponential form is methodical and simple. To make this conversion, start by identifying the base, exponent, and result from the logarithmic equation. For instance, in the equation \( \log_{10} 100,000 = 5 \), the components are:

• Base: 10
• Exponent: 5
• Result: 100,000

With these components, the exponential form is constructed by placing the base and exponent on one side of the equation and the result on the other. In this case, it becomes \( 10^{5} = 100,000 \). This tells you that 10, the base, raised to the power of 5, the exponent, equals 100,000, the result.

By converting to exponential form, the relationship between these numbers becomes clearer, allowing for easy verification of accuracy by calculating the exponentiation directly.