Problem 568 - Precalculus

Question

Rewrite \(\log _{t}(96)-\log _{t}(8)\) in compact form.

Step-by-Step Solution

Answer

\(\log _{t}(12)\)

Step 1: Understanding Logarithmic Properties

To solve this problem, we need to use the properties of logarithms. Specifically, the property \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right).\)This means that the difference of two logs with the same base is the log of the quotient of their arguments. We will use this property to rewrite \(\log _{t}(96)-\log _{t}(8).\)

Step 2: Applying the Quotient Rule

Using the quotient rule, we can rewrite \(\log _{t}(96)-\log _{t}(8)\)as follows:\[\log _{t}(96) - \log _{t}(8) = \log _{t}\left(\frac{96}{8}\right).\]Next, we will simplify inside the logarithm by dividing the numbers 96 and 8.

Step 3: Simplifying the Expression Inside the Logarithm

Perform the division inside the logarithm: \(\frac{96}{8} = 12.\)Now the expression becomes:\(\log _{t}(12).\)This is the simplified compact form of the original logarithmic expression.

Key Concepts

Logarithmic Identities

Logarithmic identities are mathematical tools that help us manipulate and simplify logarithmic expressions. These identities are derived from the fundamental properties of logarithms, which are based on exponential functions. One commonly used identity is the "difference of logs" rule:

• \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \)

This identity allows us to combine two separate logarithmic expressions into one by transforming the difference of logs into the log of a quotient. When logs have the same base, this rule ensures we can streamline expressions for easier calculation. Applying these identities is crucial for simplifying complex logarithmic statements and solving equations involving logs. Recognizing when to use these properties enhances problem-solving efficiency and deepens understanding of logarithmic relationships.
In the exercise, we used this identity to turn \( \log _{t}(96) - \log _{t}(8) \) into a single logarithm \( \log _{t}\left(\frac{96}{8}\right) \).

Quotient Rule

The quotient rule for logarithms is a fundamental concept in simplifying logarithmic expressions. It provides a direct way to handle the division of terms within logarithms. The rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator:

• \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)

This property is particularly useful when simplifying expressions where two logarithms with the same base are subtracted. Using the quotient rule helps transition from a potentially more complex form to a single, concise logarithm.
In practice, when we see an expression like \( \log_t(96) - \log_t(8) \), the quotient rule guides us to rewrite it as a single logarithm \( \log_t\left(\frac{96}{8}\right) \). This simplifies the problem and makes further calculations straightforward by focusing on a single entity, rather than multiple components.

Simplification of Expressions

Simplification of expressions is a common goal in algebra and higher-level math. It involves reducing expressions to their simplest forms by using various mathematical rules and identities. In the context of logarithms, simplification often follows the application of basic logarithmic properties or identities, such as the product, quotient, or power rules.
After applying the logarithmic identities, the next step in simplifying \( \log_t\left(\frac{96}{8}\right) \) is to simplify the fraction \( \frac{96}{8} \). By performing straightforward division, we find the quotient to be 12, thus turning the expression into \( \log_t(12) \).
This final form, \( \log_t(12) \), is much more compact and manageable, clearly illustrating the power of simplification in reducing the complexity of mathematical expressions. By recognizing opportunities to simplify, we can make calculations more efficient and our understanding of the problems more profound.