Problem 47 - College Algebra Essentials

Question

Use properties of logarithms to condense 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 condensed and simplified single logarithm expression for \(\log (2x+5) - \log x\) is \(\log (2+(5/x))\).

Step 1: Identify the Property

The logarithm property that needs to be applied here is the quotient rule of logarithms. This rule says that subtracting two logs with the same base equals the log of the quotient of what the logs were of. So, \(\log_b a - \log_b c\) can be rewritten as \(\log_b (a/c)\). The base 'b' can be any positive number, except 1.

Step 2: Apply the Quotient Rule

We apply the quotient rule to the given expression \(\log (2x+5) - \log x\). Here a is (2x+5) and c is x. So, the expression becomes \(\log ((2x+5)/x)\).

Step 3: Simplify the Expression

After applying the properties of logarithms, the expression can be further simplified. So, \(\log ((2x+5)/x)\) simplifies to \(\log (2+(5/x))\).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a helpful tool when dealing with the subtraction of two logarithms that have the same base. It states that the difference of two logs is equivalent to a single logarithm of the division of their arguments. In formula terms, it looks like this:

• If you have two logs, \( \log_b a - \log_b c \),
• this is equal to \( \log_b (\frac{a}{c}) \).

This rule is very useful because it allows you to combine two separate logarithms into a single expression.
When applying the quotient rule on the expression \( \log(2x+5) - \log x \), we let \( a = 2x+5 \) and \( c = x \). Substituting into the quotient rule gives us \( \log((2x+5)/x) \). This reformation simplifies the process of handling logarithmic expressions and will make further calculations easier.
Understanding this rule will make solving logarithmic problems less daunting and much more straightforward.

Simplifying Logarithmic Expressions

Once we've applied the quotient rule of logarithms to combine the logs, it's essential to simplify the resulting expression. Simplification involves reducing the expressions into a more compact or understandable form.
After using the quotient rule on \( \log((2x+5)/x) \), it can be simplified to \( \log(2 + \frac{5}{x}) \). This step is about making the argument of the logarithm as simple as possible.

• The key point here is to perform basic algebraic operations within the argument when possible.
• This includes simplifying fractions or combining like terms.

Such simplification is crucial because it forms expressions that are easier to understand and solve in subsequent steps.
Ultimately, simplifying expressions improves clarity and eases further manipulation involving the logarithm.

Condensing Logarithms

Condensing logarithms means taking a longer expression with multiple logarithms and writing it as a single, simpler logarithm. This process utilizes the properties of logarithms, such as the quotient rule, to bring an expression to its most compact form.
In the given exercise, we initially had \( \log(2x+5) - \log x \). By applying the quotient rule, we condensed it into the simplified log expression of \( \log(2 + \frac{5}{x}) \). This is as condensed as possible because it's been expressed as a single logarithmic function.

• The main advantage of condensing logarithms is that it simplifies complex problems and makes calculations computationally easier.
• It is particularly useful in solving equations where simplifying the terms can lead to clearer, more direct solutions.

Condensing reduces clutter, enhances understanding, and allows one to focus on solving the main problem without unnecessary complication.