Problem 45 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log \frac{10 x}{y}\)

Step-by-Step Solution

Answer

\(1 + \log x - \log y\)

Step 1: Identify the Logarithmic Rule

We will use the logarithmic rule that the logarithm of a quotient is equal to the difference of the logarithms: \[\log \left( \frac{a}{b} \right) = \log a - \log b\]

Step 2: Apply the Logarithm Rule for Quotients

Apply this rule to \(\log \frac{10x}{y}\) by breaking it into two separate logarithms:\[\log \frac{10x}{y} = \log (10x) - \log y\]

Step 3: Identify the Logarithmic Rule for Products

Next, we'll use the logarithmic rule that the logarithm of a product is equal to the sum of the logarithms:\[\log (ab) = \log a + \log b\]

Step 4: Apply the Logarithm Rule for Products

Apply this rule to \(\log (10x)\):\[\log (10x) = \log 10 + \log x\]

Step 5: Combine All Parts Together

Substitute \(\log (10x)\) back into the equation from Step 2:\[\log \frac{10x}{y} = (\log 10 + \log x) - \log y\]

Step 6: Simplify the Expression

We know that \(\log 10 = 1\) because 10 is the base of the common logarithm. Substitute this value in:\[(\log 10 + \log x) - \log y = (1 + \log x) - \log y\]This simplifies to:\[1 + \log x - \log y\]

Key Concepts

Understanding Logarithm Rules

Logarithm rules are essential tools for simplifying complex logarithmic expressions. These rules provide straightforward mechanisms to break down logarithmic terms into more manageable parts. Here are some basic rules:

• Product Rule: The logarithm of a product is the sum of the logarithms of the factors. Mathematically, it is expressed as \( \log (ab) = \log a + \log b \).
• Quotient Rule: The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. This is written as \( \log \left( \frac{a}{b} \right) = \log a - \log b \).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base: \( \log (a^b) = b \cdot \log a \).

These rules allow you to decompose logs into sums and differences, making calculations easier. We used these principles in our problem to transform and simplify the expressions.

Exploring Logarithmic Identities

Logarithmic identities help in understanding the properties and transformations of logarithms. These identities play a crucial role in both simplifying logarithmic expressions and solving equations involving logarithms. Some key identities include:

• The Identity Logarithm: \( \log 1 = 0 \) for any base, because any number raised to the power of zero is one.
• Log of the Base: \( \log_b b = 1 \), since a base raised to the power of one equals itself.
• Inverse Logarithms: When the argument of a logarithm equals the base, the result is the exponent: \( b^{\log_b x} = x \).

These identities, coupled with the rules, enable us to manipulate logarithmic expressions seamlessly, turning complex quotients into simpler differences.
In our specific problem, recognizing that \( \log 10 = 1 \) was instrumental to getting the solution simplified significantly.

Techniques for Simplifying Logarithms

Simplifying logarithms involves using logarithmic rules and identities to express logs in a simpler form. It helps make complex expressions easier to work with. Here’s how you can simplify logarithms step-by-step:1. **Break It Down**
Use the logarithmic rules to break down large expressions into smaller, simpler parts. For instance, \( \log \left( \frac{10x}{y} \right) \) gets broken into \( \log(10x) - \log(y) \) using the quotient rule.2. **Apply the Product Rule**
For terms like \( \log(10x) \), apply the product rule: \( \log(10) + \log(x) \). This converts products into the sum of logarithms.
3. **Simplify using Identities**
Substitute known values such as \( \log 10 = 1 \) to simplify the expression further. Here, \( 1 + \log x - \log y \) is the refined result.
Each step builds on the last, leveraging rules and identities to transform the original expression into a more digestible format. This not only aids in comprehension but also in solving more challenging problems effectively.