Problem 72 - Algebra 2

Question

Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient. $$ \log _{4} 3 x $$

Step-by-Step Solution

Answer

\(\log_{4}3x = \frac{\log 3x}{\log 4}\)

Step 1: Identify Base and Argument

Inspect the expression \(\log_{4}3x\). Our base is 4 and our argument is \(3x\).

Step 2: Apply Base Conversion Formula

Use the base conversion formula, \(\log_b a = \frac{\log_c a}{\log_c b}\), where \(a\) is the argument of our original log (which is \(3x\) in this case), \(b\) is the base of our original log (which is 4), and \(c\) is any positive value different from 1. Here, we interchange \(b\) with 10 to make it a common log. Our formula then becomes \(\log_{4}3x = \frac{\log 3x}{\log 4}\).

Step 3: Express Final Answer

The expression cannot be simplified any further as per the condition mentioned in the problem. So, the given logarithm expressed as the quotient of two common logarithms is \(\frac{\log 3x}{\log 4}\).

Key Concepts

Common Logarithms

Common logarithms have a base of 10 and are typically denoted as simply \( \log \) without a subscript. This detail makes them particularly useful in various calculations since the base 10 aligns with our decimal system. Often, calculators are programmed to compute common logarithms directly, making them easily accessible for everyday use.
In the context of the exercise, expressing a logarithm in terms of common logarithms allows for easier computation and interpretation. Our task was to rewrite \( \log_{4}3x \) using common logarithms. This involves utilizing the base conversion formula to transform our original expression into one that uses base 10, simplifying further calculations.

Base Conversion Formula

The base conversion formula is a tool that helps us convert logarithms from one base to another. This formula is written as:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

where \( a \) is the argument, \( b \) is the original base of the logarithm, and \( c \) is the new base we aim to convert to, often 10 in the case of common logarithms.
To give more insight, in our example with \( \log_{4}3x \), we chose the common logarithm as the base \( c \). Implementing the formula helps in converting any logarithmic expression into a more universally understood and standardized form, using common logarithms. It allows for easier application in both scientific and engineering contexts due to the standardized base.

Logarithmic Expressions

Logarithmic expressions involve numbers or variables under a logarithmic operation. They provide a way to articulate numbers in relation to their exponents of a specific base.
Understanding the parts of a logarithmic expression is crucial: it consists of an argument, which is the value you wish to find the log of, and the base, which is the number that the argument is relative to. For example, in \( \log_4 3x \), 3x is the argument, while 4 is the base.
In simplifying or rewriting logarithmic expressions, like converting them to common logs as in our exercise, we aim to use rules like the base conversion formula. This can effectively express logarithmic expressions in different, often more manageable, forms without changing their inherent values. It's an essential skill to maneuver through complex mathematical problems with ease, providing clarity in how we represent and handle numerical relationships in logarithmic functions.