Problem 100 - 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.$$3 \ln e^{5}$$.

Step-by-Step Solution

Answer

The exact value of the given logarithm is 15.

Step 1: Apply Logarithmic Property

The logarithmic power rule states that the exponent to which the base of the logarithm is raised can be brought down to the front as multiplication. Apply this rule to the given logarithmic expression. Thus the expression \(3 \ln e^{5}\) becomes \(3*5 \ln e\).

Step 2: Simplify the Expression

The natural logarithm of e \(\ln e\) is equal to 1. So the expression simplifies to \(3*5 * 1\) which equates to 15.

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is essential for simplifying complex expressions. One of the fundamental properties is:

• Product Property: \[ \ln(a \cdot b) = \ln a + \ln b \]
• Quotient Property: \[ \ln\left(\frac{a}{b}\right) = \ln a - \ln b \]
• Power Property: This is where the power of a term can be brought in front, which is crucial to this exercise. For any number \(a\), base \(b\), and exponent \(c\), \[ \ln(a^c) = c \cdot \ln a \]

These properties facilitate the simplification of expressions and are used extensively in calculus and higher-level mathematics. Recognizing them can help break down a problem into manageable parts.
It's important to practice these properties to understand their applications fully.

Logarithmic Power Rule

The logarithmic power rule is a handy tool that allows us to simplify expressions where the variable is an exponent. This is particularly useful when you're dealing with expressions like \( \ln(e^5) \).

• According to this rule, the exponent can be taken outside the logarithm as a multiplier. Thus, \[ \ln(e^5) = 5 \cdot \ln e \]
• After applying the power rule, the problem gets simplified significantly, making it easier to handle without a calculator.

This rule is derived from the basic properties of logarithms and helps in reducing the complexity of mathematical expressions. Once you understand this, solving expressions becomes much more intuitive and faster. Learning how to apply this rule will enhance your mathematical skill set, especially in calculus.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions involves multiple steps that use the properties and rules of logarithms. Let's see how we can approach it:

• Simplifying \(\ln e\): The natural logarithm of \(e\), denoted as \(\ln e\), is always equal to 1. This is because \(e\) raised to the power of 1 is \(e\) itself.\[ \ln e = 1 \]
• Multiply and Simplify: In our specific problem, \(3 \ln e^5\), we applied the power rule first, which gives us \[ 3 \times 5 \cdot \ln e \].
• Substituting \(\ln e = 1\), we get:\[ 3 \times 5 \times 1 = 15 \]

By following these steps, we reduce the expression to a simple numerical form. Efficient simplification requires a solid understanding of these properties and rules, enabling seamless calculations without depending on a calculator.