Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the common logarithmic expression without using a calculator. \(\log (0.001)\)

Step-by-Step Solution

Answer

The value of \( \log(0.001) \) is \(-3\).

Step 1: Understand the Logarithmic Function

The common logarithm, denoted as \( \log(x) \), generally represents \( \log_{10}(x) \). This means that \( \log(0.001) \) asks us to find the exponent of 10 that results in 0.001.

Step 2: Rewrite the Number in Exponential Form

Express 0.001 as a power of 10. Since 0.001 can be written as \( 10^{-3} \), we have: \[ 0.001 = 10^{-3} \]

Step 3: Apply the Logarithm Identity

Using the identity \( \log_{10}(10^x) = x \), we apply it to our expression: \[ \log(10^{-3}) = -3 \]

Step 4: Conclude the Calculation

Based on the identity applied, the value of \( \log(0.001) \) simplifies to \( -3 \). Therefore, \[ \log(0.001) = -3 \].

Key Concepts

Common Logarithm

The common logarithm is a specific type of logarithm that uses 10 as its base. It is denoted by \( \log(x) \) and it essentially answers the question: "What power do we need to raise 10 to, in order to get \( x \)?"

• For example, \( \log(100) = 2 \) because \( 10^2 = 100 \).
• In our given exercise, we are asked to find \( \log(0.001) \). This means we are looking for which power of 10 equals 0.001.

This concept is foundational in logarithms because the base of 10 is widely used in the decimal system, making it one of the most common logarithms encountered in everyday math. Learning to identify and use common logarithms helps in solving equations and real-world problems without a calculator.

Exponential Form

Understanding exponential form is crucial to solving log problems. Exponential form allows us to express large or small numbers as powers of a base number.

• For instance, the number 1000 can be written as \( 10^3 \), meaning "10 raised to the power of 3."
• Conversely, 0.001 can be expressed in exponential form as \( 10^{-3} \), indicating "10 raised to the power of -3."

Rewriting numbers in exponential form is especially useful when dealing with logarithms. Once we have the number expressed as a power of the base, finding the log becomes a straightforward process of identifying the exponent. This method simplifies the calculation and helps in understanding the relationship between log functions and exponentiation.

Logarithm Identity

Logarithm identities are important tools that simplify complex calculations. One significant identity is that the logarithm of a power is equal to the exponent itself: \( \log_{10}(10^x) = x \).

• This identity directly allows us to simplify expressions where the number is clearly a power of 10.
• For example, if we have \( 10^{-3} \), since the base of the logarithm is also 10, we can immediately say that \( \log(10^{-3}) = -3 \).

This identity eradicates the need for lengthy calculations, especially when you have an expression like \( \log(x) \) where \( x \) is a power of 10 or can be rewritten in that manner. Mastering this identity enhances your ability to deal effortlessly with logarithmic expressions.