Problem 29 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\ln e^{4.5}$$

Step-by-Step Solution

Answer

The exact value of \(\ln e^{4.5}\) is 4.5.

Step 1: Recognize the log identity

There is a log property that states \(\log_b b^a = a\). This applies equally to natural logarithms (where the base is \(e\)), i.e., \(\ln e^a = a\). In this case, \(a\) is 4.5.

Step 2: Apply the log identity

Substitute \(a\) with 4.5 in the equation \(\ln e^a = a\), we get \(\ln e^{4.5} = 4.5\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( ln \), is a specific type of logarithm that has the constant \(e\) as its base. The constant \(e\) is approximately equal to 2.71828 and is an irrational number, meaning it cannot be expressed as a simple fraction. The natural logarithm is commonly used in various fields such as mathematics, physics, and engineering due to its natural properties in calculus and exponential growth models.

Understanding the natural logarithm involves recognizing its function. When you see \( ln(x)\), it means "the power to which you must raise \(e\) to obtain \(x\)." For instance, if \(\ln(x) = 2\), then \(e^2 = x\). When calculating natural logarithms, the base \(e\) often simplifies processes, especially when dealing with exponential equations.

Logarithmic Identity

A logarithmic identity is a rule that applies to logarithms, providing a way to simplify logarithmic expressions. A fundamental identity to remember is \(\log_b b^a = a\), which also holds for natural logarithms as \(\ln e^a = a\). This identity states that when the base of the logarithm and the base of the exponent are the same, the expression simplifies directly to the exponent itself.

In the exercise \(\ln e^{4.5}\), applying this identity yields the solution directly. This is because \(\ln e^{4.5}\) simplifies to 4.5 according to the rule \(\ln e^a = a\). Recognizing and applying this identity can greatly simplify solving logarithmic problems without the use of a calculator. This is one of the reasons why logarithmic identities are powerful tools in mathematics.

Properties of Logarithms

The properties of logarithms are essential rules that aid in the manipulation and simplification of logarithmic expressions. Here are some key properties:

• Product Property: \(\log_b (xy) = \log_b x + \log_b y\).
• Quotient Property: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\).
• Power Property: \(\log_b (x^a) = a\log_b x\).
• Change of Base Formula: \(\log_b x = \frac{\log_k x}{\log_k b}\).

Understanding these properties can transform complex logarithmic expressions into simpler ones. For instance, knowing the power property allows us to simplify the expression \(\ln e^{4.5}\) by recognizing it as \(4.5\ln e\), which further simplifies to 4.5 because \(\ln e = 1\). These properties are particularly useful when solving logarithmic equations or when graphing logarithmic functions.