Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$\log (3 x+7)-\log x$$

Step-by-Step Solution

Answer

\(\log (3x + 7) - \log x = \log (3 + \frac{7}{x})\).

Step 1: Apply the Logarithm Rule

Use the logarithm property \(\log a - \log b = \log \frac{a}{b}\) to combine the two logarithms into one: \(\log (3x + 7) - \log x = \log \frac{(3x+7)}{x}\)

Step 2: Simplify the Expression

Notice that \(x\) in the denominator cancels out with \(x\) in the numerator: \(\log \frac{(3x+7)}{x} = \log (3 + \frac{7}{x})\). The expression has now been expressed as a single logarithm with a coefficient of \(1 \).

Step 3: Evaluate the Logarithm

Further simplification or evaluation of the logarithm is not possible unless specific numerical value(s) for \(x\) is provided. So the final answer remains \(\log (3 + \frac{7}{x})\).

Key Concepts

Condensing Logarithmic Expressions

In mathematics, condensing logarithmic expressions involves combining multiple logarithms into a single expression. This often makes calculations simpler and easier to manage. The key rule to remember is the difference rule of logarithms:

• When you subtract two logs with the same base, such as \(\log a - \log b\), you can combine them into a single log by dividing the arguments, resulting in \(\log \frac{a}{b}\).

For example, given \(\log (3x + 7) - \log x\), you would identify the individual logs as having a common base.
This allows you to apply the rule, converting it into a single logarithmic expression: \(\log \frac{(3x+7)}{x}\).
This process not only condenses the expression but also lays the groundwork for further algebraic simplification.

Logarithmic Identities

Logarithmic identities are fundamental tools that make managing and simplifying logarithmic expressions possible. They are built on the properties of logarithms and include:

• Product Rule: \(\log a + \log b = \log (a \cdot b)\)
• Quotient Rule: \(\log a - \log b = \log \frac{a}{b}\)
• Power Rule: \(\log a^b = b \cdot \log a\)

Using these rules correctly enables us to process expressions more effectively.
For example, the exercise uses the Quotient Rule to combine \(\log (3x+7)\) and \(\log x\).
These identities are essential because they reduce complex logarithmic relationships into simpler forms that are easier to compute or analyze.

Algebraic Manipulation

Algebraic manipulation is a critical skill in mathematics that involves simplifying expressions using algebraic rules and operations. When working with logarithmic expressions, we often need to simplify the result further or evaluate it.
In the given example, after applying the logarithm rules to condense the expression to \(\log \frac{(3x+7)}{x}\), a simple algebraic step is used to remove common factors.
This results in an expression that breaks it down to \(\log (3 + \frac{7}{x})\), which can't be simplified further without specific values for \(x\).
In cases like this, understanding how to manipulate and simplify algebraic structures is essential, as it helps in transitioning seamlessly from one mathematical approach to another.

• Mastering basic algebraic skills ensures that no matter the complexity of the initial expression, one can shift it into a more manageable form.

Algebraic manipulation, combined with logarithmic identities, provides a powerful toolkit for handling complex calculations.