Problem 5 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (1000 x) $$

Step-by-Step Solution

Answer

\(\log(1000x) = \log(1000) + \log(x) = 3 + \log(x)\)

Step 1: Apply the Logarithm Property

Apply the property of logarithm of a product to \(\log(1000x)\), which means we can express it as the sum of \(\log(1000)\) and \(\log(x)\). So, \(\log(1000x) = \log(1000) + \log(x)\).

Step 2: Evaluate Logarithm

Evaluate logarithmic expressions where possible without using a calculator. So \(\log(1000)=3\), because the base of the logarithm is 10. The equation now becomes \(3 + \log(x)\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve the logarithm function, which is the inverse of exponentiation. Understanding how to work with logarithms is a fundamental skill in algebra and higher mathematics. When we see an expression like \( \text{log}(1000x) \), we are looking at a logarithmic expression where \( \text{log} \) is commonly understood to mean the base-10 logarithm.

Manipulating logarithmic expressions typically involves using a set of specific properties of logarithms, which enable us to expand, condense, or evaluate expressions. These properties are based on the core concepts of logarithms. For example, when we expand \( \text{log}(1000x) \), we are really asking, what exponent do we need to raise 10 to, to get \( 1000x \)? This is the core of what a logarithm represents, and knowing this helps us understand how to manipulate and evaluate logarithmic expressions properly.

Expanding Logarithms

Expanding logarithms is the process of using logarithmic properties to rewrite a single logarithmic expression into a sum or difference of logarithms. This is often done to simplify the expression or to solve logarithmic equations. In our exercise, we used the property that a logarithm of a product can be written as the sum of the logarithms of the factors.

So, \( \text{log}(1000x) \) expands to \( \text{log}(1000) + \text{log}(x) \) using the property of logarithms: \( \text{log}(ab) = \text{log}(a) + \text{log}(b) \). Being adept at expanding logarithms is crucial for further operations in mathematics such as integration, differentiation, and solving complex equations. It's important for students to practice this skill to improve their fluency in working through logarithmic problems.

Evaluating Logarithms

Evaluating logarithms means finding the numeric value of a logarithmic expression, usually without the aid of a calculator. It requires an understanding of the nature of logarithms and the relationship between exponentiation and logarithms.

In our example, evaluating \( \text{log}(1000) \) involves understanding that since \( 10^3 = 1000 \), the logarithm of 1000 with a base of 10 is 3. Thus, \( \text{log}(1000) \) evaluates to 3. What this means is that the power we must raise 10 to, in order to get 1000, is simply 3. This simplifies our initial expression to \( 3 + \text{log}(x) \). When learning to evaluate logarithms, it's useful to memorize the logarithms of commonly used bases, such as 10 and e, to simplify calculations and improve speed and accuracy in problem-solving.