Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} 10 x$$.

Step-by-Step Solution

Answer

The expanded form of \( \log _{10} 10x \) is 1 + \( \log_{10}x \)

Step 1: Apply the Product Rule

Start by applying the product rule which states that the logarithm of a product is the sum of the logarithms of the factors. That converts \( \log _{10} 10x \) into \( \log_{10}10 + \log_{10}x \).

Step 2: Simplify Logarithm Base 10

Now simplify \( \log_{10}10 \) as it's a fundamental logarithmic property that states the log of a number with the same number as its base equals 1. So, \( \log_{10}10 \) simplifies to 1.

Key Concepts

Product Rule

The product rule for logarithms is a fundamental concept that simplifies complex logarithmic expressions into easier parts. When you have the logarithm of a product, the product rule allows you to express it as the sum of the logarithms of its factors.

In mathematical terms, for two positive numbers \(a\) and \(b\):

• \(\log_b (a \cdot c) = \log_b a + \log_b c\)

This transformation is particularly useful in breaking down multiplicative combinations under a logarithmic expression. With the original exercise, we can see this in action with \(\log_{10} (10x)\). Applying the product rule, this can be expanded to \(\log_{10} 10 + \log_{10} x\), neatly separating the components of the product.

This rule not only simplifies calculations but also provides clearer insights when dealing with logarithmic equations involving products.

Logarithmic Expansion

Logarithmic expansion is the process of breaking down a single logarithmic expression into multiple simpler logs. This involves using properties like the product rule, quotient rule, and power rule to deconstruct complex logarithmic expressions. Each log component corresponds to a part of the original expression, detailed by specific operations—multiplication, division, or exponentiation.

By expanding \(\log_{10} (10x)\) as \(\log_{10} 10 + \log_{10} x\), we see each part contributes to the whole.

Logarithmic expansion is particularly helpful in solving equations or derivative functions in calculus, as it makes the calculations straightforward. This breakdown allows students to see how each factor within the product relates to its logarithmic expression, heightening understanding through clear separation of terms.

• Begins with applying and understanding basic properties like the product rule.
• Helps in rationalizing complex expressions and solving them efficiently.

Simplifying Logarithms

Simplifying logarithms often makes use of basic logarithmic identities and properties. The core idea is to reduce the expression to its simplest form, making it easier to work with or understand.

A basic fundamental property that aids in this is the understanding that the logarithm of a number with its base is 1, such as \(\log_{10}10 = 1\). This concept stems from the fact that 10 raised to the power of 1 gives 10.

When performing logarithmic expansions, like in our example, recognizing these simplifications plays a crucial role. Once the product rule \(\log_{10} (10x) = \log_{10}10 + \log_{10}x\) is applied, it becomes easy to identify and simplify to \(1 + \log_{10} x\).

Simplifying logarithms reduces potential computation errors and speeds up algebraic and calculus-based problem-solving processes. It's an essential part of using logarithms effectively.