Problem 34 - 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.) $$2 \ln e^{6}-\ln e^{5}$$

Step-by-Step Solution

Answer

\nThe exact value of the logarithmic expression \(2 \ln e^{6} - \ln e^{5}\) without using a calculator is 17.

Step 1: Apply the logarithm power rule

The power rule of logarithms allows you to move the exponent of the argument of a logarithm out front as a coefficient. Applying this, we have: \(2 \ln e^{6}\) becomes \(12\), and \(-\ln e^{5}\) becomes \(-5\). The power rule is applied because the base of the natural log, \(e\), raised to any power \(x\), equals \(x\), i.e. \(\ln e^{x} = x\).

Step 2: Perform the subtraction

Now that we have two constant values, perform the subtraction to get the final answer: \(12 - (-5) = 12 + 5 = 17\).

Step 3: State the solution

The exact value of the logarithmic expression \(2 \ln e^{6} - \ln e^{5}\) without using a calculator is 17.

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them easier to manipulate and solve in complex expressions. These properties can simplify calculations and reveal insights that are not immediately obvious. Here are the main properties you need to know:

• Product Rule: This rule states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, this is expressed as \( \log_b (xy) = \log_b x + \log_b y \).
• Quotient Rule: According to this rule, the logarithm of a quotient is the difference of the logarithms. It is represented by \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \).
• Power Rule: The power rule states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. It is given by \( \log_b (x^y) = y \cdot \log_b x \).

These properties are incredibly useful, especially when dealing with large numbers or complex logarithmic expressions. They allow us to break down problems into smaller, more manageable parts.

Natural Logarithm

The natural logarithm, often abbreviated as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). The constant \( e \) is the base of natural logs and is crucial in many areas of mathematics and natural sciences.

Natural logarithms have some unique properties:

• They simplify exponential expressions involving the base \( e \). This is because \( \ln e^x = x \), making them particularly useful in calculus for solving differential and integral equations.
• The natural logarithm of 1 is always 0, since \( e^0 = 1 \).
• Natural logs are used extensively in growth/decay models, like population growth, radioactive decay, and in financial calculations involving continuous compounding interest.

Understanding natural logarithms is key to solving mathematical problems involving exponential growth and decay, and they frequently appear in higher-level mathematics.

Logarithm Power Rule

The logarithm power rule is one of the essential tools for simplifying expressions with exponents inside a logarithm. The power rule states that when you have a logarithm of a number raised to a power, you can "bring down" the exponent to be a multiplier of the logarithm.

This is expressed as \( \log_b (x^y) = y \cdot \log_b x \). This property makes solving logarithmic questions simpler, as it allows us to transform complex expressions into linear forms, which are generally easier to handle.

In practical applications, this rule is key when dealing with exponential decay or growth models. It also frequently appears in equations where you are solving for an unknown that is an exponent, especially in calculus and algebra.

By understanding and applying the power rule, you can break down and simplify even the most daunting tasks involving logarithms, moving from complexity to clarity.