Problem 47 - College Algebra

Question

In Exercises \(41-70,\) 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 without using a calculator. $$ \log (2 x+5)-\log x $$

Step-by-Step Solution

Answer

The final answer after condensing the logarithmic expression \( \log (2x+5) - \log x \) would be \( \log(2 + 5/x) \)

Step 1 - Identify the logarithm property

Recognize here we have a subtraction between two logarithms. Remember the properties of logarithms, particularly this one: \( \log_b a - \log_b c = \log_b (a/c) \). This equation represents that the subtraction of logs equals to the log of the division of the arguments.

Step 2 - Apply the logarithm property

Apply the property to our expression \( \log (2x+5) - \log x \). According to the property, this is equivalent to \( \log ((2x+5) / x) \).

Step 3 - Simplify the expression

Simplify the expression inside the log as much as possible. Divide (2x+5) by x to simplify further: \( \log ((2x/x) + (5/x)) \), which simplifies to \( \log(2 + 5/x) \).

Key Concepts

Condensing Logarithms

Condensing logarithms involves combining multiple logarithmic terms into a single term. This process uses the properties of logarithms to simplify expressions effectively. One essential property for condensing involves subtraction of logs, which can be rewritten as the division of their respective arguments. This particular property is mathematically expressed as:

• \( \log_b a - \log_b c = \log_b \left( \frac{a}{c} \right) \)

The goal of condensing is to reduce the complexity of the logarithmic expression. By applying these properties, expressions such as \( \log (2x+5) - \log x \) turn into a single expression \( \log \left( \frac{2x+5}{x} \right) \). This makes the expression compact and sometimes easier to evaluate or further manipulate.

Logarithm Subtraction

Subtraction in logarithms can often seem tricky, but it's a process founded on the idea of division. When you have a subtraction of two logarithmic expressions, you can transform it into a single logarithm by dividing the arguments of those logarithms.

• For example, \( \log (2x+5) - \log x \).
• According to the property, this is equivalent to \( \log \left( \frac{2x+5}{x} \right) \).

This transforms two separate log operations into one, simplifying the expression by consolidating it. Understanding this concept is crucial for working with logs, as it often makes mathematical computation and simplification more straightforward.

Logarithmic Expressions Simplification

Simplifying logarithmic expressions is a key skill in algebra, aimed at making complex expressions easier to handle. After condensing logarithms into a single term, the next step is simplifying the argument itself where possible. This involves basic algebraic manipulation such as distributing, factoring, or canceling terms.

• For the expression \( \log \left( \frac{2x+5}{x} \right) \), a simplification step could be:

• Rewriting it as \( \log \left( \frac{2x}{x} + \frac{5}{x} \right) \), which simplifies to \( \log (2 + \frac{5}{x}) \).

Such simplification helps in understanding the core structure of the mathematical relationship being expressed. By breaking down complex expressions into simpler forms, one can solve equations more efficiently and gain better insight into the behavior of mathematical functions.