Problem 30 - Contemporary Precalculus

Question

Evaluate the given expression without using a calculator. $$\ln \sqrt[5]{e}$$

Step-by-Step Solution

Answer

Answer: $$\frac{1}{5}$$

Step 1: Recall the properties of logarithms

To simplify the expression, we need to remember two important properties of logarithms: 1. The natural logarithm of e is 1: $$\ln(e) = 1$$ 2. The logarithm of a root can be expressed as a fraction: $$\log_a \sqrt[b]{x} = \frac{1}{b} \log_a x$$

Step 2: Apply the logarithm of a root property

We can now apply the second property to rewrite the given expression: $$\ln \sqrt[5]{e} = \frac{1}{5} \ln{e}$$

Step 3: Use the natural logarithm property

Now using the natural logarithm of e, which is equal to 1, we substitute it into the expression: $$\frac{1}{5} \ln{e} = \frac{1}{5}(1)$$

Step 4: Evaluate the expression

Finally, we can evaluate the expression and find the answer: $$\frac{1}{5}(1) = \frac{1}{5}$$ The value of $$\ln \sqrt[5]{e}$$ is $$\frac{1}{5}$$.

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm. It is denoted by \(\ln\), and it refers to a logarithm with base \(e\). The number \(e\) is approximately equal to 2.718 and is an important constant in mathematics. The natural logarithm is widely used in calculus and complex mathematics.

• When \(x = e\), the natural logarithm yields: \(\ln(e) = 1\).
• This property simplifies calculations involving exponential growth or decay.

Understanding the natural logarithm is essential when dealing with equations that involve the exponential constant, \(e\). It is frequently found in problems related to growth rates in fields like finance, physics, and biology. Be familiar with its properties to simplify expressions and solve equations effectively.

Properties of Logarithms

Logarithms have several properties that make complex calculations more manageable. These properties help in simplifying expressions by transforming them into a form where operations become straightforward and easier.

• Product Rule: \(\ln(a \times b) = \ln(a) + \ln(b)\).
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).
• Power Rule: \(\ln(a^b) = b \times \ln(a)\).
• Root as a fraction: \(\ln(\sqrt[b]{a}) = \frac{1}{b} \times \ln(a)\).

These rules are useful when simplifying logarithmic expressions. For the expression \(\ln \sqrt[5]{e}\), we used the property of a root as a fraction to simplify initially. Each property can be helpful depending on the situation, so it's good to keep them in mind whenever dealing with logarithms.

Exponents

Exponents are a way to symbolize repeated multiplication of a number by itself. In mathematics, exponents play a crucial role especially in simplifying expressions and solving equations.

• An exponent indicates how many times the base number is multiplied by itself: \(a^n = a \times a \times \ldots \times a\) \(n\) times.
• Exponential functions include powers such as squares, cubes, and roots.

In the context of logarithms, the inverse relationship between logs and exponents is very useful. This relationship allows us to convert multiplication into addition, division into subtraction, powers into multiplication, and roots into division using the properties of logarithms. Understanding exponents is essential for exploring how logarithms work with various mathematical expressions easily.

Simplifying Expressions

Simplifying expressions in mathematics means reducing them to their simplest form. This process often involves applying rules and properties to make equations easier to understand and solve.

• Identify the parts of the expression that can be simplified using known properties (like those of logarithms and exponents).
• Look for opportunities to apply specific rules, like combining like terms or using logarithm properties.
• Rewriting complex terms into simpler ones often makes solving the problem much more accessible.

In the original problem, we simplified \(\ln \sqrt[5]{e}\) by recognizing it could be expressed as \(\frac{1}{5} \ln e\). Then, knowing \(\ln e = 1\), we arrived at the final simplified answer of \(\frac{1}{5}\). Mastering simplification techniques enables us to tackle even more complex mathematical challenges effectively.