Problem 16 - 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. $$ \log _{10} 100,000=5 $$

Step-by-Step Solution

Answer

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

Step 1: Identify Components

In the given logarithmic expression, \( \log_{10} 100,000 = 5 \), identify the base, the result, and the exponent. Here, the base is 10, the result is 100,000, and the exponent is 5.

Step 2: Convert to Exponential Form

To convert the logarithmic expression to exponential form, use the relationship between logarithms and exponents, which is: if \( \log_{b}(a) = c \), then \( b^{c} = a \). Applying this to \( \log_{10} 100,000 = 5 \), we write \( 10^{5} = 100,000 \).

Step 3: Verify Conversion

Double-check the converted expression by computing \( 10^{5} \) to ensure it equals 100,000. Calculate \( 10 \times 10 \times 10 \times 10 \times 10 = 100,000 \), confirming the conversion is correct.

Key Concepts

Exponential Form

When we talk about exponential form, we're referring to a way of expressing numbers as powers of a base. In this form, you have a base raised to a certain exponent or power. This mathematical expression is often represented as \( b^c = a \).
It shows that the base \( b \), when multiplied by itself \( c \) times, results in the number \( a \).
Exponential form is a concise way of writing high-value numbers. For example, instead of writing out 100,000 as a simple number, you could express it as \( 10^5 \) in exponential form.

• Base: the number that is multiplied.
• Exponent: shows how many times the base is used as a factor.
• Result: the final product of multiplying the base by itself exponent times.

Understanding exponential form is fundamental in math and science, as it provides a consistent way to handle large numbers.

Base and Exponent

In any exponential expression, the base and exponent play crucial roles. The base is the number that is repeatedly multiplied, while the exponent indicates how many times the base is used in the multiplication.
For instance, in \( 2^3 \), 2 is the base, and 3 is the exponent, which means 2 is multiplied by itself three times:
\[ 2 \times 2 \times 2 = 8 \]

• Base: The number being multiplied.
• Exponent: Shows the power to which the base is raised (repeated multiplication).
• Expression: The entire statement \( b^c \) is called a power.

Recognizing the base and exponent helps in converting logarithmic expressions into their exponential form, thus simplifying their use in calculations.

Logarithmic to Exponential Conversion

Logarithms and exponentials are closely related concepts. The conversion process allows you to switch between logarithmic and exponential forms, making calculations more manageable in certain contexts.
When you see a logarithmic statement like \( \log_b(a) = c \), it tells you that \( b^c = a \). In other words, the logarithm calculates the exponent that the base must be raised to, in order to get the result.
Here's how to do the conversion:

• Identify the base \( b \), the result \( a \), and the logarithm's value \( c \).
• Rewrite the statement in exponential form: \( b^c = a \).
• Verify by calculating \( b^c \). If your calculation results in \( a \), your conversion is correct.

For example, converting \( \log_{10} 100,000 = 5 \) to its exponential form gives \( 10^5 = 100,000 \), confirming the value of the exponent in the original logarithm.