Problem 66 - 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: Express the Logs in terms of Base 10

Convert each logarithm to common logarithmic form, which is base 10, using the change of base formula: \[ \log_{b}a = \frac{\log_{10}a}{\log_{10}b} \]So we have:1. \( \log_{2}5 = \frac{\log_{10}5}{\log_{10}2} \)2. \( \log_{5}7 = \frac{\log_{10}7}{\log_{10}5} \)

Step 2: Multiply the Expressions

Combine the expressions from Step 1. Multiplying them gives:\[ \left( \frac{\log_{10}5}{\log_{10}2} \right) \left( \frac{\log_{10}7}{\log_{10}5} \right) \]

Step 3: Simplify the Multiplication

Notice that \( \log_{10}5 \) appears in both numerator and denominator. Cancel these terms:\[ \frac{\log_{10}5 \times \log_{10}7}{\log_{10}2 \times \log_{10}5} = \frac{\log_{10}7}{\log_{10}2} \]

Step 4: Convert Back to Logarithms

Using the change of base formula in reverse, convert back to logarithm of a different base:\[ \frac{\log_{10}7}{\log_{10}2} = \log_{2}7 \]

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in logarithmic calculations. Sometimes, you're faced with logarithms that are not of a familiar base. If you know only common logarithms (base 10) or natural logarithms (base e), you can convert any logarithm to these bases using this formula:
\[\log_{b}a = \frac{\log_{10}a}{\log_{10}b}\]
This means if you have a logarithm of base \(b\), you can express it using common logarithms. This is especially useful when simplifying complex products or divisions of log expressions where log tables or calculators typically provide values only for base 10 or base e.
You can use this formula not just to switch to base 10 but to convert any base \(a\) logarithm to a base \(c\) logarithm by modifying the formula to:

• \(\log_{b}a = \frac{\log_{c}a}{\log_{c}b}\)

This universal applicability makes the change of base formula a staple in working with logarithms.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves using properties that allow us to make expressions less complex. One common technique is using the product, quotient, and power rules, but in this exercise, we closely examined utilizing the change of base formula followed by simplification.
The initial step involves converting each logarithmic term to a common base, like base 10, to make multiplication or other operations more straightforward. For example, converting \(\log_2 5\) and \(\log_5 7\) into base 10 forms simplifies the expression:

• \(\log_2 5 = \frac{\log_{10}5}{\log_{10}2}\)
• \(\log_5 7 = \frac{\log_{10}7}{\log_{10}5}\)

Once you have comparable terms, you can combine and simplify them. By multiplying and identifying common terms in the numerator and denominator, we can cancel them out, further simplifying our expression. This simplification is a great example of how understanding base conversions assists in efficiently reducing complexity in calculations.

Logarithmic Identities

Understanding and applying logarithmic identities can make tricky logarithmic expressions much simpler. In the provided exercise, several basic identities were used:

• Identity Property: \(\log_a a = 1\). This is foundational because it implies that a number to the power of itself is 1.


• Inverse Property: \(\log_a 1 = 0\), meaning that any number's power to reach 1 is 0.
• Product to Sum: High-level log manipulations often involve converting the product of logs into a sum, but it wasn’t directly used in the original exercise.

Identity recognition is crucial in simplifying, reducing terms, and ensuring accuracy in manipulating logs. Being able to identify and cancel \(\log_{10}5\) in both the numerator and the denominator is part of leveraging these identities. This technique often opens doors to recognizing further simplifications, as we saw when deriving \(\log_2 7\). These identities provide the tools necessary for working seamlessly across different mathematical problems involving logarithms.