Problem 12 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{5}\left(\frac{125}{y}\right)\)

Step-by-Step Solution

Answer

The expression \(\log_{5}(\frac{125}{y})\) can be expanded as \(3 - \log_{5}(y)\)

Step 1: Identify the property to use

The given logarithmic expression consists of a quotient, \(\frac{125}{y}\). The property of logarithm states that the logarithm of a quotient is the difference of the logarithms. i.e. \(\log{\frac{a}{b}} = \log{a} - \log{b}\).

Step 2: Apply the Quotient Logarithm Rule to Expand the Expression

By applying the quotient rule, the given expression \(\log_{5}(\frac{125}{y})\) can be written as \(\log_{5}(125) - \log_{5}(y)\).

Step 3: Simplify the Expanded Expression

The value of \(\log_{5}(125)\) is 3 because \(5^3 = 125\). Therefore, the expanded logarithmic expression simplifies to \(3 - \log_{5}(y)\).

Key Concepts

Logarithm of a Quotient

Understanding the logarithm of a quotient is crucial because it helps us to simplify complex logarithmic expressions. According to this property, the logarithm of a fraction can be expressed as the difference of two separate logarithms. Specifically, the formula is given by:

• \(\log{\frac{a}{b}} = \log{a} - \log{b}\)

In our original exercise, we have the expression \(\log_{5}\left(\frac{125}{y}\right)\). By employing this logarithmic property, we can rewrite this as \(\log_{5}(125) - \log_{5}(y)\).
This step makes it easier to evaluate or further simplify the expression because you've broken it down into simpler components. Always remember, breaking down a problem into smaller, easily solvable parts is a powerful strategy in mathematics.

Expand Logarithmic Expressions

The process of expanding logarithmic expressions involves using logarithmic properties to transform these expressions into a more workable form. This often means rewriting a single logarithmic term into multiple terms, which is particularly useful for simplifying or solving the expression.
Let's consider the term \(\log_{5}(\frac{125}{y})\). By understanding that this expression follows the form of the logarithm of a quotient, as discussed earlier, we know that it expands to \(\log_{5}(125) - \log_{5}(y)\).

• Expanding in this manner helps reveal opportunities for further calculations or simplifications, like evaluating any numerical logarithms or combining like terms.
• This process is vital for anyone working with logarithms, as it lays down the basic groundwork for more complex problem-solving tasks.

Once expanded, you can then check if any parts of the expanded expression can be evaluated with simple arithmetic.

Evaluate Logarithms Without Calculator

There are times when you may need to evaluate logarithmic expressions without having the luxury of a calculator. Understanding how to do this is essential for solving problems quickly and accurately. Let’s explore this using the example \(\log_{5}(125)\).
To evaluate this, recall the definition of a logarithm: it is the exponent to which a base must be raised to yield a specific number.
Thus, when you see \(\log_{5}(125)\), you need to find the power \(x\) such that

• \(5^x = 125\).
• By evaluating the powers of 5, you may notice that \(5^3 = 125\).

Therefore, \(\log_{5}(125) = 3\). This step is critical because it reduces the complexity of our expanded expression to \(3 - \log_{5}(y)\). With practice, recognizing such logarithmic values becomes quicker, allowing for efficient problem-solving.