Problem 60 - Algebra and Trigonometry

Question

Simplify: \(\left(\log _{2} 5\right)\left(\log _{5} 7\right)\)

Step-by-Step Solution

Answer

The expression simplifies to \(\log_2 7\).

Step 1: Apply Change of Base Formula

We'll first use the change of base formula for logarithms: \ \[ \log_b a = \frac{\log_c a}{\log_c b} \] \ Let's apply this to both logarithms in the expression \((\log_2 5)(\log_5 7)\). We can choose the natural logarithm (ln) for simplification: \ \[ \log_2 5 = \frac{\ln 5}{\ln 2} \] \ \[ \log_5 7 = \frac{\ln 7}{\ln 5} \]

Step 2: Multiply the Expressions

Substitute the expressions from Step 1 into the original expression: \ \[ \left(\frac{\ln 5}{\ln 2}\right) \left(\frac{\ln 7}{\ln 5}\right) \] \ Now, multiply these expressions. Notice that \(\ln 5\) appears in both the numerator of the first fraction and the denominator of the second fraction, allowing us to cancel it out: \ \[ \frac{\ln 5 \cdot \ln 7}{\ln 2 \cdot \ln 5} = \frac{\ln 7}{\ln 2} \]

Step 3: Express as a Single Logarithm

The expression \(\frac{\ln 7}{\ln 2}\) is equivalent to \(\log_2 7\), using the change of base formula backwards. Therefore, the simplified form of the original expression \((\log_2 5)(\log_5 7)\) is \(\log_2 7\).

Key Concepts

Change of Base Formula

The change of base formula is a handy tool when working with logarithms of different bases. Imagine you are working with logs with bases other than 10 or e (Euler's number). The change of base formula allows you to convert any logarithm to a different base that might be more convenient for your calculations.
\[\log_b a = \frac{\log_c a}{\log_c b} \]
This formula states that you can express the log of a number `a` with base `b` using any other base `c`. Simply divide the logarithm of `a` with base `c` by the logarithm of `b` with base `c`.

• Why use it? Changing to a base like 10 or e allows you to use calculators more effectively.
• Helps simplify complex expressions involving multiple logarithms of different bases.

In our exercise, we used the natural logarithm (base e) to apply the change of base formula for easier calculation and cancellation.

Simplification of Logarithms

Simplifying logarithmic expressions might look overwhelming at first, but with the right tools, it becomes straightforward. The simplification often involves using algebraic rules and properties to make the expression easier to understand and compute. Here's how it works in our problem:
First, we used the change of base formula to rewrite both terms \((\log_2 5)(\log_5 7)\). This turned them into fractions using natural logarithms: \[ \log_2 5 = \frac{\ln 5}{\ln 2} \]
\[ \log_5 7 = \frac{\ln 7}{\ln 5} \] Rewriting expressions this way is a crucial step as it lays the foundation for further simplification.
Once both logarithms were rewritten, it allowed us to multiply and cancel terms in the fractions, specifically canceling out the common \(\ln 5\). This led to a simpler expression: \( \frac{\ln 7}{\ln 2} \).
Remember:

• Simplification can often involve looking for terms to cancel or combining logarithms into single expressions.
• Always be mindful of maintaining equivalence when rewriting or simplifying expressions.

Properties of Logarithms

Logarithms have several essential properties that make them powerful tools for solving equations and simplifying expressions. Understanding these properties can turn a daunting logarithmic problem into a simple arithmetic exercise.

• Product Rule: \( \log_b (xy) = \log_b x + \log_b y \)
• Quotient Rule: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• Power Rule: \( \log_b (x^y) = y \cdot \log_b x \)
• Change of Base Formula: \( \log_b a = \frac{\log_c a}{\log_c b} \)

These properties help in transforming and simplifying expressions, as seen in our problem. By applying these rules intelligently, you can rework complex problems into much simpler forms.
In our exercise, we used the change of base formula and then canceled terms using properties of fractions. Recognizing opportunities to apply these logarithmic properties will always lead to easier calculations and clarity in your solutions.