Problem 68 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\log _{5} 8-\log _{5} t$$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \(\log_{5} \frac{8}{t}\).

Step 1 - Identifying the rule of logarithms to apply

Here we can identify that the given expression is in the form of a difference of two logarithms with the same base, i.e., \(\log _{5} 8 - \log _{5} t\). This qualifies for the application of the quotient rule of logarithms.

Step 2 - Apply the Quotient rule

We know the quotient rule of logarithms, according to which \(\log_b{\frac{a}{b}} = \log_b{a} - \log_b{b}\). The given expression falls under this rule, which implies that we can write it as: \(\log_{5} \frac{8}{t}\)

Step 3 - Simplifying the Expression

The expression \(\log_{5} \frac{8}{t}\) is already in its simplest form, as it is a single logarithm.

Key Concepts

Logarithmic Expressions

In mathematics, logarithmic expressions represent equations that involve the logarithm of a number. Logarithms can be visualized as the inverse operation of exponentiation. When working with logarithmic expressions, it's essential to understand the base of the logarithm. For example, in the expression \( \log_5 8\), the base is 5.

• The base tells us which power will result in a specific number. For instance, \(\log_5 8\) asks '5 raised to what power equals 8?'
• A logarithmic expression can involve various operations such as addition, subtraction, multiplication, or division typically associated with specific rules.

Understanding these operations is crucial for manipulating and simplifying logarithmic expressions effectively.

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential tool for simplifying expressions where two logarithms are subtracted. This rule states:

• \(\log_b{\frac{M}{N}} = \log_b{M} - \log_b{N}\)

This means when you have the subtraction of two logarithms with the same base, it can be condensed into a single logarithm representing the division of their arguments.
For instance, given \(\log_5 8 - \log_5 t\), we can apply the quotient rule and rewrite it as \(\log_5 \frac{8}{t}\).
It's a powerful rule since it allows the combination of multiple logarithms into one, making calculations simpler and more manageable.

Simplifying Logarithms

When we talk about simplifying logarithms, we mean reducing the given expression to its most compact form, usually a single logarithm. Once we apply rules like the quotient rule, the expression \(\log_5 \frac{8}{t}\) is considered simplified.

• Simplifying logarithms often involves recognizing patterns or rules that apply, such as the product, quotient, or power rule.
• The goal is to transform complex expressions into straightforward and concise forms for easier interpretation and calculation.

By mastering these simplification techniques, students can tackle various algebraic problems involving logarithms with confidence and efficiency.