Problem 42 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each 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 given logarithmic expression \( \log 250 + \log 4 \) simplifies to a single logarithm \( \log 1000 \) with the coefficient \( 1 \), which equals \( 3 \).

Step 1: Apply the Logarithmic Addition Property

The property of logarithms that will be used is the addition property: \( \log_b mn = \log_b m + \log_b n \), if \( b \), \( m \), and \( n \) are positive and \( b \ne 1 \). Applying this property to the given expression we get: \[ \log 250 + \log 4 = \log (250 \times 4) \]

Step 2: Perform Mathematical Operation

The next step will be to multiply \( 250 \) and \( 4 \) to simplify the expression. \[ \log (250 \times 4) = \log 1000 \]

Step 3: Simplify Logarithmic Expression

Knowing that \( \log 1000 \) to the base \( 10 \) equals \( 3 \), this expression simplifies to: \[ \log 1000 = 3 \]

Key Concepts

Logarithmic Addition Property

The logarithmic addition property is an essential tool that helps simplify the combination of logarithms. It states that for any real numbers where the base is not equal to 1 and all involved numbers are positive, the logarithm of the product is equal to the sum of the logarithms. In simple terms, the property can be expressed as: \( \log_b (mn) = \log_b m + \log_b n \). So, when you are given an expression like \( \log 250 + \log 4 \), you can combine these into a single logarithmic expression. By applying the addition property, you turn this into \( \log (250 \times 4) \). This reduction maintains the integrity of the expression while representing it more concisely.

Condensing Logarithmic Expressions

Condensing logarithmic expressions essentially means combining multiple logarithmic terms into a single term. This process often uses properties of logarithms like the addition property. The goal is to take several terms and transform them into one. For example, using our exercise, starting with \( \log 250 + \log 4 \), we use the logarithmic addition property to rewrite it as \( \log (250 \times 4) \). By multiplying the numbers inside the logarithm, we get a condensed form — \( \log 1000 \), which is a single term rather than two separate ones. Condensation makes it easier to manage calculations, especially when you're applying other operations like simplification.

Simplifying Logarithms

Simplifying logarithms involves reducing them to their most basic form. Once a logarithmic expression is condensed, occasionally, it can be further simplified to a numeric value. For instance, once condensing the expression results in \( \log 1000 \), simplification is the next logical step. Since 1000 is the base 10 raised to the power of 3, \( \log 1000 = \log_{10} (10^3) \), this simplifies to \( 3 \). This streamlined result occurs because the base 10 logarithm tells us the power to which 10 must be raised to produce the number under the log. Simplification can be a critical step for solving equations efficiently.

Base 10 Logarithm

The base 10 logarithm, often known as the common logarithm, is one of the most used forms of logarithms. It is denoted either as \( \log(x) \) or \( \log_{10}(x) \) and is prevalent in scientific calculations due to its close relation to exponential growth and scientific notation. Base 10 logarithms are convenient because they reflect the power of 10 necessary to reach a given number. Using our example, \( \log 1000 \) implies determining what power 10 must be raised to result in 1000. The answer is 3, since \( 10^3 = 1000 \). Understanding base 10 logarithms is crucial for various mathematical and scientific contexts, allowing quick interpretation of large and small values.