Problem 19 - 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. $$ \log _{10} 0.001=-3 $$

Step-by-Step Solution

Answer

The exponential form is \( 10^{-3} = 0.001 \).

Step 1: Identify Logarithmic Statement

The given logarithmic statement is \( \log_{10} 0.001 = -3 \). This represents a logarithm with base 10, solving for exponent \(-3\) resulting in 0.001.

Step 2: Understanding Logarithmic Form

The logarithmic form \( \log_b a = c \) signifies that \( b^c = a \). Here, \( b \) is the base, \( a \) is the resulting value, and \( c \) is the exponent.

Step 3: Rewrite in Exponential Form

Based on the logarithmic identity \( \log_b a = c \equiv b^c = a \), we can rewrite \( \log_{10} 0.001 = -3 \) as \( 10^{-3} = 0.001 \).

Key Concepts

Logarithmic Form

The logarithmic form is a way to express exponentiation using a special structure. When you see a statement such as \( \log_b a = c \), it's telling you that the base \( b \), when raised to the exponent \( c \), gives you the number \( a \). Think of logarithms as the opposite of exponents. They help answer the question: "To what power should I raise the base \( b \) to get \( a \)?"

• "Base" is the number we are repeatedly multiplying.
• "Result" is what we get after the multiplication.
• "Exponent" is how many times we multiply the base.

It's really useful when you want to solve for the exponent in exponential equations. So, whenever you see a problem in logarithmic form, try to recognize the base, exponent, and result.

Exponential Form

Rewriting a logarithmic expression in exponential form helps us see the relationship as a multiplication problem. For example, if you have \( \log_{10} 0.001 = -3 \), you can convert this to exponential form as \( 10^{-3} = 0.001 \). Here are some points to remember:

• The "base" \( b \) remains the same.
• The "exponent" \( c \) tells us how many times the base is used as a factor.
• The "result" \( a \) is the outcome of raising the base to the power given by the exponent.

You can think of this conversion as a way to show the behind-the-scenes workings of the logarithm. Each part of the logarithmic statement finds its place in the exponential form, making it easier to calculate and understand.

Logarithmic Identity

A logarithmic identity like \( \log_b a = c \equiv b^c = a \) is a useful tool in mathematics, essentially defining the relationship between logarithms and exponents. Recognizing this identity is crucial across various math problems, as it provides a reliable way to translate logarithmic expressions into exponential ones.

Key Features of Logarithmic Identity:

• The identity highlights the equivalence between a logarithmic statement and its exponential counterpart.
• It serves as a bridge, making it straightforward to move from one format to another.
• Understanding it can aid in solutions of equations and simplifications.

Whether you're solving complex equations or simplifying expressions, the logarithmic identity is a foundational concept. It gives clarity to the interaction between logarithms and exponents, allowing for seamless transitions and greater comprehension of mathematical principles.