Problem 36 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln 4-1$$

Step-by-Step Solution

Answer

The simplified expression in the form of single logarithm is \(\ln \frac{4}{e}\)

Step 1: Apply logarithm properties

The first step is to apply the stated property of logarithms to simplify the given expression. According to the properties of logarithms, the difference between two logarithms of the same base can be written as a single logarithm by dividing the arguments of the logarithms. Therefore, \(\ln 4 - 1 = \ln 4 - \ln e\) since the natural logarithm has a base of \(e\), and therefore 1 can be written as \(\ln e\).

Step 2: Rewrite as a single logarithm

The expression from step 1 can now be rewritten as a single logarithm, using the property of logarithms that \(\log_a m - \log_a n = \log_a \frac{m}{n}\). When this property is applied to the expression from step 1, it gives \(\ln 4 - \ln e = \ln \frac{4}{e}\).

Step 3: Simplify the expression

The expression \(\ln \frac{4}{e}\) is already in its simplest form as a single logarithm, as theres no further simplification possible without knowing the exact value of \(e\).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools for manipulating and simplifying logarithmic expressions. These properties include several rules that allow us to combine, break down, or transform logarithmic terms:

• Product Rule: This rule states that the logarithm of a product is the sum of logarithms: \[ \log_a (mn) = \log_a m + \log_a n \] This is useful when you have two numbers multiplied together and want to express them as log terms.
• Quotient Rule: The logarithm of a quotient is the difference between the logarithms: \[ \log_a \left( \frac{m}{n} \right) = \log_a m - \log_a n \] This helps when dividing numbers and wanting to express them as log terms, as seen in our exercise where \( \ln 4 - \ln e = \ln \left( \frac{4}{e} \right) \).
• Power Rule: The logarithm of a number raised to a power can be expressed by multiplying the power by the logarithm of the base number: \[ \log_a (m^n) = n \cdot \log_a m \].

Using these properties, we can convert complex logarithmic expressions into simple single logarithms, making our calculations easier to handle.

Natural Logarithm

The natural logarithm is a specific type of logarithm that has the base of the irrational number \(e\), where \(e \approx 2.71828\). We denote natural logarithms with \(\ln\) instead of the usual \(\log\). Natural logarithms frequently occur in science and engineering due to their close relationship with continuous growth and decaying processes.In our exercise, distinguishing that \(1\) can be written as \(\ln e\), is crucial for simplifying the expression. This is because the natural logarithm of its base, \(e\), is equal to 1, i.e., \(\ln e = 1\). This property aids in expressing numbers like \(1\) in terms of ln, making it possible to apply logarithmic properties to combine expressions efficiently.

Expression Simplification

Expression simplification involves reducing a mathematical expression to its simplest form. Simplifying expressions helps make them easier to interpret and can often reveal important insights or solutions. In logarithmic simplification:- Start with identifying terms that can be combined using logarithmic properties. In the exercise, identify the terms \(\ln 4\) and \(1\). Recognize that \(1\) can be rewritten as \(\ln e\), which allows us to use the quotient rule.- Use the logarithmic quotient rule: Here, the expression \(\ln 4 - 1\) can be rewritten as \(\ln 4 - \ln e\), and further simplified using the quotient property to become \(\ln \left( \frac{4}{e} \right)\).In summary, the simplification of the expression relies on recognizing patterns and properties within logarithms to consolidate into a single, more manageable log term. This also underscores the importance of understanding logarithmic identities and being able to recognize how to apply them to reduce complex expressions.