Problem 11 - College Algebra

Question

For the following exercises, condense to a single logarithm if possible. $$ \log _{b}(28)-\log _{b}(7) $$

Step-by-Step Solution

Answer

\( \log_{b}(4) \)

Step 1: Identify Logarithmic Property

We recognize that we have a subtraction of two logarithms with the same base, which suggests using a logarithmic property that can simplify this expression.

Step 2: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that the difference of two logarithms with the same base can be written as the logarithm of the quotient of their arguments. Here, apply the rule: \( \log_{b}(28) - \log_{b}(7) = \log_{b}\left(\frac{28}{7}\right) \).

Step 3: Simplify the Quotient

Now we simplify the expression inside the logarithm. Calculate \( \frac{28}{7} = 4 \). This simplifies our expression to \( \log_{b}(4) \).

Key Concepts

Quotient Rule of Logarithms

Understanding logarithmic properties can make complex expressions simpler. One of the fundamental properties is the Quotient Rule of Logarithms. This rule helps us manage situations where we have subtraction involving logs that share the same base. In simple terms, when you see two logarithms subtracted like this: \( \log_{b}(A) - \log_{b}(B) \), you can use the Quotient Rule to transform this into a single logarithm: \( \log_{b}\left(\frac{A}{B}\right) \).
This property is a handy tool, especially when dealing with equations that have many logarithms because it helps consolidate and reduce the number of terms you’re working with.
By converting the subtraction of two logarithms into a single logarithm, calculations become more manageable and less prone to error. It's important to always check that the base of the logarithms is the same before applying this rule.

Simplifying Expressions

After converting the subtraction of logarithms into a single log using the Quotient Rule, your next step is to simplify the expression inside the logarithm. In our example, the expression becomes \( \log_{b}\left(\frac{28}{7}\right) \).
The task now is to carry out the division \( \frac{28}{7} \), which would simplify to \( 4 \). This process of simplifying the fraction is crucial because it reduces the complexity of the equation you work with.
Simplifying expressions also helps in better understanding the behavior of the function and determining the value more quickly. Always look to perform any available simplifications as each reduces unnecessary complexities in your problem-solving process.

Subtraction of Logarithms

Subtraction in logarithms might seem tricky at first, but it's straightforward once you understand the underlying rules. The subtraction of logarithms, as addressed in this solution, leads us directly to make use of properties like the Quotient Rule.
In logarithmic terms, when you subtract, you are essentially dividing their arguments. Thus, \( \log_{b}(28) - \log_{b}(7) \) simplifies because it can be interpreted as dividing 28 by 7, which is why it turns into \( \log_{b}(4) \) after applying the rule and simplifying.
Remember that subtracting logs isn't about removing the terms but transforming them into a simpler equivalent, easing the flow of log calculations. This transformation is crucial in both solving equations and understanding more advanced logarithmic concepts.