Problem 77 - Precalculus

Question

Prove the quotient rule: \(\log _{b}\left(\frac{M}{N}\right)=\log _{b} M-\log _{b} N\) Hint: Let \(u=\log _{b} M\) and \(v=\log _{b} N .\) Write both in exponential form and find the quotient \(\log _{b}\left(\frac{M}{N}\right)\)

Step-by-Step Solution

Answer

The proof shows that \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\) using properties of logarithms and exponents.

Step 1: Define Variables in Exponential Form

We are given the variables: let \(u = \log_b M\) and \(v = \log_b N\). By definition of logarithm, the exponential form is \(M = b^u\) and \(N = b^v\).

Step 2: Set Up the Quotient in Logarithmic Terms

We need to find \(\log_b\left(\frac{M}{N}\right)\). Use the exponential expressions for \(M\) and \(N\) to rewrite this as: \(\log_b\left(\frac{b^u}{b^v}\right)\).

Step 3: Apply Exponent Quotient Rule

Whenever you divide like bases, you subtract the exponents: \(\frac{b^u}{b^v} = b^{u-v}\). Thus, we have: \(\log_b\left(b^{u-v}\right)\).

Step 4: Use Logarithmic Identity

By the property of logarithms, \(\log_b(b^{u-v}) = u - v\). This leads to: \(u - v = \log_b M - \log_b N\).

Step 5: Conclude the Proof

Therefore, by substituting back the original variables, we confirm that \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\). We have just proven the quotient rule for logarithms.

Key Concepts

Logarithmic Identities

Logarithmic identities are key formulas that simplify working with logarithms. These identities help us manipulate logarithmic expressions, just like rules in algebra. Here, we'll focus on proving the quotient rule. This rule states that the logarithm of a quotient is the difference of the logarithms:

• \( \log_b\left( \frac{M}{N} \right) = \log_b M - \log_b N \)

This identity comes from the fundamental property of logarithms being the inverse of exponentiation. It aligns with the idea that division in logarithms translates to subtraction in their values. Understanding this identity is crucial for solving more complex problems that involve logarithmic expressions. In simpler terms, when you see a division inside a log, think subtraction!

Exponential Form

Exponential form is a way to express numbers as powers of a base. When working with logarithms, translating between logarithmic and exponential forms is crucial. This can make complex equations more approachable:

• If \( u = \log_b M \), then \( M = b^u \)
• If \( v = \log_b N \), then \( N = b^v \)

Using exponential form helps simplify the process of proving the quotient rule. Thus, we rewrite \( \log_b(\frac{M}{N}) \) as \( \log_b(\frac{b^u}{b^v}) \). This gives a clearer pathway to use laws of exponents to reach our conclusion. Such transformations are the backbone of transitioning between different mathematical frameworks, making complex mathematical ideas easier to digest.

Properties of Logarithms

The properties of logarithms are powerful tools that can reduce the complexity of calculations. One such property is the behavior of logarithms with powers and roots. Let's highlight some foundational properties:

• Product Rule: \( \log_b(MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b(M^k) = k \log_b M \)

In our proof, we specifically used the quotient rule. By recognizing \( \log_b(b^{u-v}) = u-v \), we highlight how dividing powers leads to subtracting logarithms. Fully grasping these properties allows for efficient simplification and manipulation of logarithmic expressions, which is essential in many areas of higher mathematics.