Problem 38 - Calculus and its applications

Question

Given \(\ln 4=1.3863\) and \(\ln 5=1.6094,\) find each value. Do not use a calculator. $$ \ln \sqrt{e^{6}} $$

Step-by-Step Solution

Answer

\(\ln \sqrt{e^{6}} = 3\).

Step 1: Simplifying the Square Root

The expression given is \(\ln \sqrt{e^{6}}\). Recall that the square root of a number is the same as raising that number to the power of \(\frac{1}{2}\). Thus, \(\sqrt{e^{6}} = (e^{6})^{\frac{1}{2}}\). Simplifying this expression, we get \(e^{3}\).

Step 2: Applying Logarithmic Identity

We need to find \(\ln(e^{3})\). By applying the logarithmic identity \( \ln(a^b) = b \ln(a)\), we can simplify this to \(3 \ln(e)\).

Step 3: Using the Natural Logarithm of e

Recall that the natural logarithm of \(e\) is 1, i.e., \(\ln(e) = 1\). Therefore, \(3 \ln(e) = 3 \times 1\).

Step 4: Final Calculation

After simplifying, we obtain \(3 \times 1 = 3\). Thus, \(\ln \sqrt{e^{6}} = 3\).

Key Concepts

Logarithmic Identities

Logarithmic identities are essential tools in making complex logarithmic problems simpler and easier to solve. These identities relate the operations of logarithms to exponential functions and are based on the properties of powers and roots.
One of the most critical identities is the power rule for logarithms, which tells us that \(\ln(a^b) = b \ln(a)\). This means that we can take the exponent and multiply it by the logarithm. It's a simple yet powerful transformation.
Another useful identity is the natural logarithm of the base \(e\), which is a mathematical constant approximately equal to 2.71828. The identity states that \(\ln(e)=1\). This property simplifies calculations significantly, especially when dealing with exponential functions involving \(e\).
Logarithmic identities help us in rearranging and breaking down expressions to find solutions without complicated calculations. They form the basis for simplifying log equations and make solving problems involving natural logarithms more intuitive.

Exponential Functions

Exponential functions are functions of the form \(f(x) = a^x\), where \(a\) is a positive constant, and \(x\) is the variable. These functions have unique properties and behaviors that make them important in mathematics and sciences.
One special case of an exponential function involves the natural base \(e\). Here, the function is written as \(f(x) = e^x\). The natural base \(e\) appearing in exponential functions is common in growth models, such as population and compound interest calculations.
When working with exponents, understanding their properties is key. For example:

• The product of powers: \(a^b \times a^c = a^{b+c}\).
• The power of a power: \((a^b)^c = a^{b \times c}\).

In our exercise, we used the concept of the power of a power in simplifying \(\sqrt{e^{6}}\) to \(e^3\). Recognizing these properties allows us to manipulate and simplify expressions effectively.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more basic or concise form without changing their value. This process often uses a combination of arithmetic operations and mathematical rules to reach a simpler equivalent.
It is a valuable skill because it helps reduce complex expressions into a more manageable form, making calculations easier. In mathematics, simplification often involves uncovering hidden structures or patterns, like recognizing a common logarithmic identity or exponential rule.
In our step-by-step solution, simplification began by identifying \(\sqrt{e^{6}}\) as \((e^{6})^{\frac{1}{2}}\), leading to \(e^3\). Then we applied the logarithmic identity to find \(\ln(e^3) = 3\ln(e)\).
This process helps to understand and solve the original expression \(\ln \sqrt{e^{6}}\) in a straightforward manner. Mastery of techniques for simplifying expressions is crucial in both algebra and calculus, providing clarity and precision in solutions.