Problem 23 - Algebra 2

Question

Expand each logarithm. \(\log _{5} \frac{r}{s}\)

Step-by-Step Solution

Answer

\(\log _{5} \frac{r}{s} = \log_5 r - \log_5 s\)

Step 1: Identify the Rule to be Used

Identify the quotient rule of logarithms: \(\log_b \frac{m}{n} = \log_b m - \log_b n\). In this case, \(b=5\), \(m=r\), and \(n=s\).

Step 2: Apply the Rule

Apply the quotient rule to \(\log _{5} \frac{r}{s}\). This gives us \(\log_5 r - \log_5 s\).

Key Concepts

Quotient Rule of Logarithms

In logarithmic operations, the quotient rule of logarithms is a crucial concept to understand. This rule is particularly useful when dealing with the logarithm of a quotient. The quotient rule states that the logarithm of a quotient \(rac{m}{n}\) can be expanded into the difference of two logarithms: \(\log_b \frac{m}{n} = \log_b m - \log_b n\). This allows us to break down complex logarithmic expressions into simpler, more manageable parts. By using the quotient rule, you can simplify expressions significantly faster, which is especially handy in calculus and algebra.

• The rule helps decompose fractions within a logarithm.
• It makes complex expressions easier to manipulate and solve.

Understanding this rule not only helps in expansion but also in simplifying expressions when reversing the process.

Logarithmic Expansion

Logarithmic expansion involves breaking down a compound logarithmic expression into simpler components. By expanding logarithms, you can convert complex expressions into a sum or difference of simpler logarithms. For example, with \(\log_b \frac{r}{s}\), applying the quotient rule, it expands to \(\log_b r - \log_b s\). Each part of the expanded form is simpler to handle separately.

• It makes mathematical manipulation more straightforward.
• Expansion can reveal hidden relationships within a problem.

Through expansion, computations and problem-solving become more intuitive and less prone to errors.

Base of a Logarithm

The base of a logarithm is a fundamental concept in understanding logarithmic equations. It is represented by the subscript in the expression \(\log_b x\). The base determines the value that the logarithm scales to in its calculations. For example, in \(\log_5 r\), the base is 5. This means that the logarithm measures the power to which the base, 5, must be raised to produce the corresponding number, r.

• Common bases include 10 (common logarithms) and e (natural logarithms).
• A different base changes the scale and output of calculations significantly.

Understanding the base is crucial in ensuring correct applications of logarithmic rules in diverse mathematical contexts.

Simplifying Logarithms

Simplifying logarithms is an essential skill. It includes the use of rules like the quotient rule to transform complex logarithmic expressions into simpler forms.
The process of simplifying involves using known logarithmic identities and manipulations to reduce the expression's complexity.
For instance, \(\log_5 \frac{r}{s}\) simplifies to \(\log_5 r - \log_5 s\) using the quotient rule.

• This simplification aids in solving equations faster.
• It also helps in more efficiently analyzing functions and graphs.

Simplification is not only about making the expression shorter, but also making it more approachable for further calculations and interpretations.