Problem 42 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\log 250+\log 4\)

Step-by-Step Solution

Answer

The single logarithm of the given logarithmic expression \( \log 250 + \log 4 \) is 3

Step 1: Apply properties of logarithms

The properties of logarithms state that if \( \log a + \log b \), it can be written as \( \log (a*b) \). Therefore, the given expression \( \log 250 + \log 4 \) could be rewritten as \( \log (250 * 4) \)

Step 2: Compute the product

Calculate the product 250 * 4 which is equal to 1000. So, the single logarithm becomes \( \log 1000 \)

Step 3: Evaluate Logarithmic Expressions

Note that the problem states to not use a calculator where possible. Therefore, recall the definition of a logarithm. If \( \log_a {b} = c \), then it can be rewritten as \( a^c = b \). In our case, since the base of the logarithm is not stated, it can be assumed to be 10. So, \( 10^3 = 1000 \), therefore \( \log 1000 = 3 \)

Key Concepts

Properties of Logarithms

Logarithms have certain properties that are remarkably useful when working with expressions involving them. The most common properties that you should know about include:

• Product Property: For any positive numbers \(a\) and \(b\), \(\log(a \cdot b) = \log a + \log b\). This property allows us to add logarithmic expressions with the same base by transforming them into the logarithm of the product of those numbers.
• Quotient Property: The quotient of two numbers can also be expressed as a logarithm: \(\log(\frac{a}{b}) = \log a - \log b\).
• Power Property: Raising a number inside a logarithm can be expressed as multiplication: \(\log(a^n) = n \cdot \log a\).

Understanding these properties helps in manipulating and simplifying expressions, as seen in our problem with \(\log 250 + \log 4\), helping us combine them into a single, more simplified expression.

Condensing Logarithms

Condensing logarithms involves using the properties of logarithms to rewrite multiple logarithmic terms as a single term. This process makes the expression much simpler and can be achieved by applying the properties we just discussed.

In the exercise given, we're combining \(\log 250 + \log 4\) into one concise logarithmic expression. Using the product property of logarithms, we rewrite it as \(\log(250 \cdot 4)\).

By multiplying the numbers, which is 250 times 4, you end up with 1000. So, the expression condenses down to \(\log 1000\). With a few calculations and the right understanding of logarithmic properties, what were initially two terms become a single term.

Evaluating Logarithms

Evaluating logarithms often means finding the numerical value of a logarithmic expression without the aid of a calculator. This requires a good understanding of how logarithms work.

After condensing our expression to \(\log 1000\), the goal is to figure out what it equals. Remember that without a specified base, the common logarithm is base 10.

So, the equation \(\log_{10} 1000 = x\) translates to finding what power you must raise 10 to, to get 1000. Since \(10^3 = 1000\), we know that \(\log_{10} 1000 = 3\). Often, knowing values for common powers of 10 is helpful when evaluating logarithmic expressions.

Logarithmic Properties in Algebra

In algebra, logarithmic properties are invaluable for simplifying, solving, and evaluating expressions involving logarithms. These properties allow you to break down complex expressions into simpler parts, often revealing solutions more clearly.

When solving algebraic equations involving logarithms, you'll consistently use the properties of condensing, expanding, and evaluating. Logarithms transform multiplicative relationships into additive ones, and vice versa, which is incredibly useful when working with exponential equations.

• Simplifying Expressions: Use the properties to combine or break apart logarithms as needed.
• Solving Equations: By reversing the exponentiation through logarithms, you can find unknown variables in equations involving powers.
• Real-World Applications: Logarithms appear in exponential growth and decay, pH measurement in chemistry, and more. Understanding their algebraic properties enriches your ability to handle real-world problems involving such concepts.

These applications illustrate just how powerful and versatile the understanding of logarithmic properties can be in higher-level algebra and beyond.