Question

Expand the logarithmic expression. (See Example 2.) $$\ln \frac{x}{3 y}$$

Short answer

The expanded form of the given logarithmic expression \( \ln \frac{x}{3y} \) is \( \ln x - \ln 3 - \ln y \).

Step-by-step solution

Recognize the Logarithmic Expression

The given logarithmic expression is \( \ln \frac{x}{3y} \), which can be thought of as the logarithm of a quotient or division problem.

Apply the Quotient Rule

The quotient rule of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms. So, applying the quotient rule, we rewrite the expression as \( \ln x - \ln 3y \)

Apply the Rule of Coefficients

The rule of coefficients states that if we have a coefficient in the argument of a logarithm, we can move it to the front of the logarithm and multiply. So, the expression can be further simplified to \( \ln x - \ln 3 - \ln y \)

Key concepts

Quotient Rule of Logarithms

Understanding the quotient rule of logarithms is critical when learning how to manipulate and simplify logarithmic expressions. This rule comes into play when you are dealing with the logarithm of a division, as seen in the given exercise. The quotient rule states that for any positive real numbers a and b, where b does not equal zero, the logarithm of the division of a by b is the difference between the logarithm of a and the logarithm of b, mathematically expressed as: \[\begin{equation}\log_b \left(\frac{a}{b}\right) = \log_b a - \log_b b\end{equation}\].
For example, in the exercise, we applied this rule to the natural logarithm of a quotient: \[\begin{equation}\ln \left(\frac{x}{3y}\right) = \ln x - \ln(3y).\end{equation}\]
Being able to apply the quotient rule effectively requires practice, but once mastered, it provides a powerful tool for breaking down more complex logarithmic expressions into simpler parts.

Logarithm Properties

Logarithmic expressions follow several key properties that help in their simplification and understanding. Apart from the quotient rule we discussed previously, here are other essential properties commonly used:

• Product Rule: For positive real numbers a and b, the logarithm of a product is the sum of the logarithms: \( \log_b(ab) = \log_b a + \log_b b \).
• Power Rule: The logarithm of a number raised to an exponent is the exponent times the logarithm of the number: \( \log_b(a^c) = c\cdot\log_b a \).
• Change of Base Formula: Allows the logarithm of a number with one base to be converted to a logarithm with a different base by using: \( \log_b a = \frac{\log_c a}{\log_c b} \), where c is any positive number.
• Identity Rules: Includes \( \log_b 1 = 0 \) because any number raised to the power of zero is 1, and \( \log_b b = 1 \) because any number raised to the first power is itself.

In the provided exercise, we make use of the 'rule of coefficients', which is a part of these logarithm properties. The rule allows us to move a coefficient in front of the logarithm, effectively converting a multiplication inside the log into an addition outside of it, simplifying the expression from \( \ln 3y \) to \( \ln 3 + \ln y \). These properties enable us to properly expand and simplify logarithmic expressions efficiently.

Natural Logarithm

The natural logarithm, represented by \( \ln \), is a specific type of logarithm with the base of \( e \), where \( e \) is Euler's number, approximately equal to 2.71828. This irrational number arises naturally in mathematics and is key in various branches, including calculus, complex analysis, and certain number theory aspects. The natural logarithm has unique properties that make it a frequent subject in higher-level mathematics.
The most notable property is the derivative of the natural logarithm: \[\begin{equation}\frac{d}{dx}(\ln x) = \frac{1}{x}\end{equation}\],
which serves as the foundation for solving many problems involving growth and decay. Additionally, the inverse function of the natural logarithm is the exponential function \( e^x \), making these two deeply intertwined in mathematical theory and practical applications.
When expanding the logarithmic expression in the exercise, we deal with the natural logarithm of variables and constants. Recognizing that \( \ln \) indicates a base of \( e \) can help with understanding the context of the problem and how the natural logarithm interacts with other mathematical elements, such as coefficients and variables, in the expression.