Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln e^{6}-2 \ln e^{7}$$.

Step-by-Step Solution

Answer

The exact value of the given logarithm \( \ln e^{6}-2 \ln e^{7} \) is \( \ln(6/49) \)

Step 1: Simplify the exponentials given

Firstly, as it is universally known that \( e^n = n \), \(\ e^6 = 6 \) and \( e^7 = 7 \). So rewriting the given expression, it will become: \( \ln 6 - 2 \ln 7 \)

Step 2: Apply the Power Property of Logarithms

The power rule, which applies to logarithms in all bases, comes from the power property of exponents, it states that \( log_a{m^n}=n log_a m \). Here it can be applied on the second term to rewrite the expression: \( \ln 6 - 2 \ln 7 = \ln 6 - \ln 7^2 \)

Step 3: Apply the quotient rule

The quotient rule states that the difference of the logarithm of two numbers is the logarithm of the quotient. It's reported as \( log_b(m)- log_b(n) = log_b(m/n) \). So, for our expression it will be: \( \ln 6 - \ln 7^2 = \ln (6/7^2) = \ln(6/49) \)

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a type of logarithm that uses the number \(e\) as its base. \(e\) is an irrational and transcendental number approximately equal to 2.71828. In simpler terms, the natural logarithm of a number is the power to which \(e\) must be raised to produce that number. An integral property of natural logarithms, which is extremely handy, is that \(\ln(e) = 1\) and conversely, \(e^{\ln(x)} = x\).

This plays directly into the exercise at hand, where the student might initially make the mistake of not realizing that \(e^6\) simplifies to 6 because \(e\) to the power of natural logarithm of a number returns that number itself. However, the use of \(\ln\) over other logarithms like \(\log\) which uses 10 as a base, is preferred in certain fields like calculus due to its natural occurrence in continuous growth and decay processes.

Power Property of Logarithms

The power property of logarithms states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number, which mathematically can be expressed as \(\log_a(m^n) = n \cdot \log_a(m)\).

To apply this property in our exercise, look at the second term \(2 \ln e^7\). The power property allows us to simplify this to \(\ln (e^7)^2\) which further simplifies to \(\ln e^{14}\), and since the base of the natural logarithm is \(e\), it simplifies to \(14\). Using the power property properly can help in simplifying logarithmic expressions significantly before performing operations such as addition, subtraction, multiplication, or division.

Quotient Rule of Logarithms

The quotient rule of logarithms is a way to simplify the difference of two logarithms with the same base into a single logarithm. This rule can be expressed mathematically as \(\log_b(m) - \log_b(n) = \log_b(\frac{m}{n})\). In the context of the initial exercise, we observed \(\ln 6 - \ln (7^2)\) which, with the help of the quotient rule, could be written as \(\ln(\frac{6}{7^2})\) or \(\ln(\frac{6}{49})\).

Application of this property is crucial as it not only simplifies the expression but also represents the underlying concept of logarithms—dealing with exponents and their operations. The quotient rule is often used alongside the power rule to break down complex logarithmic expressions into more manageable forms.