Problem 10 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{1000}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \log(x) - 3.

Step 1: Identify the type of logarithmic expression

The given expression is a logarithmic operation of a fraction, which means we can apply the law of logarithms that says \log(a/b) = \log(a) - \log(b). Here, 'a' represents the numerator (\(x\)) and 'b' represents the denominator (\(1000\)).

Step 2: Apply the Logarithmic Law

Substitute the fraction \(x/1000\) with 'a/b', and apply this law of logarithms. So, it becomes: \log(a) - \log(b), which is \log(x) - \log(1000).

Step 3: Evaluate the Logarithmic expression

The \log(1000) is equivalent to 3 because 10 to the power of 3 equals 1000. So, replace \log(1000) with 3. The final expanded expression becomes \log(x) - 3.

Key Concepts

Logarithmic Expressions

Understanding logarithmic expressions is foundational to grasping the concepts of logarithms and their properties. A logarithm, at its core, tells us the power to which a number, known as the base, must be raised to produce another number. The expression \( \log_b(a) \) represents the logarithm of 'a' with base 'b'

For example, in the expression \( \log(10^3) \), the number 10 is the base, and 3 is the exponent to which base 10 must be raised to equal 1000. Therefore, \( \log(10^3) = 3 \) because \( 10^3 = 1000 \). This example illustrates a simple logarithmic expression where the answer can be determined easily without a calculator.

Law of Logarithms

The law of logarithms includes several rules that simplify logarithmic expressions and make them more manageable. Among these, some fundamental laws are:

• The Product Law: \( \log(a \cdot b) = \log(a) + \log(b) \), allowing us to separate the logarithm of a product into a sum of logarithms.
• The Quotient Law: \( \log(\frac{a}{b}) = \log(a) - \log(b) \), which lets us express the logarithm of a fraction as the difference of two logarithms.
• The Power Law: \( \log(a^n) = n \cdot \log(a) \), indicating that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base number.

These rules can be applied to expand or condense logarithmic expressions, facilitating the evaluation of logarithms without the need for a calculator. Understanding and properly applying these laws are critical for working with logarithms.

Evaluate Logarithms Without Calculator

Evaluating logarithms without a calculator relies on recognizing patterns and specific values you memorize over time. Most commonly known bases are 10, e (the natural logarithm base), and 2. For instance, with base 10, any power of 10 is easy to compute:\

\( \log(1) = 0 \) since \( 10^0 = 1 \)\
\( \log(10) = 1 \) since \( 10^1 = 10 \)\
\( \log(100) = 2 \) since \( 10^2 = 100 \)\
\( \log(1000) = 3 \) since \( 10^3 = 1000 \)\

Recognizing these patterns allows students to perform quick mental calculations. A tip to remember is that any logarithm of the form \( \log(b^x) \) where 'b' is your base and 'x' is an integer, the result is going to be that integer 'x'. The previously discussed laws of logarithms can also be used to break down complex logarithms into these basic, easily recognizable forms, enabling an evaluation without the need for technology.