Problem 5 - 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. $$ \log (1000 x) $$

Step-by-Step Solution

Answer

The expanded form of \( \log (1000x) \) is \( 3 + \log x \).

Step 1: Apply the rule of logarithm of a product

This rule allows us to break down \( \log (1000x) \) into \( \log 1000 + \log x \). Here the product inside the log function, 1000x, is separated into the sum of two separate logarithms.

Step 2: Evaluate \( \log 1000 \)

Since \( 10^3 = 1000 \), we can say \( \log 10^3 = 3 \). Because the log base 10 of 10 to the power of any number equals that number, this means \( \log 1000 = 3 \).

Step 3: Compile final result

We can piece the results from Steps 1 and 2 together yielding the fully expanded form of \( \log (1000x) \) as \( 3 + \log x \).

Key Concepts

Logarithm of a Product

Understanding the logarithm of a product is a crucial skill when dealing with logarithmic expressions. The fundamental rule here states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors. For instance, the expression \(\log(ab)\) is equal to \(\log a + \log b\).

This property stems from the definition of logarithms in relation to exponents; since multiplying numbers with the same base corresponds to adding their exponents, the reverse operation, taking a log, translates this multiplication into addition. This makes complex logarithmic expressions much more manageable.

In our example, \(\log (1000x)\) is effectively broken down into \(\log 1000 + \log x\), showcasing the property's application. When expanding logarithmic expressions, it is a powerful tool that can simplify the process significantly, making it a foundational concept to master for evaluating logarithms efficiently.

Evaluate Logarithms

Evaluating logarithms involves determining the exponent that a given base must be raised to in order to obtain a certain value. It's similar to asking, 'To what power must our base number be raised to produce this value?'.

As we saw in the exercise, \(\log 1000\) can be evaluated by recognizing that 1000 is a power of 10, specifically, \(10^3\). Hence, the logarithm base 10 of 1000 is 3, which we express as \(\log 1000 = 3\).

This method can be applied to any logarithm where the argument is a recognizable power of the base; this skill is particularly valuable as it allows for evaluating logarithms without a calculator. It's important to become familiar with the common bases and their powers to make this process more intuitive.

Logarithmic Expressions

Logarithmic expressions refer to any expression that includes logarithms. They're not just simple logs; they can be as complex as containing sums, differences, products, and quotients within a logarithm function. Understanding their properties allows us to expand, condense, or transform these expressions for various purposes, such as solving equations or simplifying calculations.

For example, the given logarithmic expression, \(\log (1000x)\), can be expanded by recognizing the product within the logarithm. After expansion, it can be further simplified if we identify parts that can be evaluated without a calculator. This simplicity can then assist further if the expression is part of a larger calculation or equation.

The natural log (ln) follows the same properties, and remembering this aids in tackling a broader range of problems. Mastery of manipulating logarithmic expressions is a key step towards a deeper understanding of logarithms as a mathematical concept.