Problem 41 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$\ln e^{\pi}$$

Step-by-Step Solution

Answer

The value is \( \pi \).

Step 1: Understand the Expression

The expression given is \( \ln e^{\pi} \). It represents the natural logarithm of \( e^{\pi} \). In mathematical terms, we need to simplify this expression using the properties of logarithms.

Step 2: Apply Logarithm Property

One key property of logarithms is that \( \ln a^b = b \ln a \). In our expression, \( a = e \) and \( b = \pi \), so we can apply this property to simplify \( \ln e^{\pi} \) to \( \pi \ln e \).

Step 3: Recall the Logarithmic Identity

Another important identity in logarithms is \( \ln e = 1 \) because \( e \) is the base of the natural logarithm. Thus, \( \pi \ln e = \pi \times 1 \).

Step 4: Simplify the Expression

Given that \( \ln e = 1 \), the expression becomes \( \pi \times 1 = \pi \). Therefore, the value of the original expression \( \ln e^{\pi} \) simplifies to \( \pi \).

Key Concepts

Properties of Logarithms

Logarithms have special properties that make them easier to manipulate and calculate. Using these properties wisely can simplify complex expressions quickly. One significant property involves powers: when you have an expression of the form \( \ln a^b \), you can bring the exponent \( b \) in front of the logarithm, rewriting it as \( b \ln a \).

• This move is useful because it changes a potentially complicated expression into a simpler multiplication.
• It leverages the idea that repeated multiplications (exponents) can be dealt with as repeated additions (logarithms).
• Therefore, applying this principle to \( \ln e^{\pi} \), the expression becomes \( \pi \ln e \).

Understanding the properties of logarithms can make evaluating expressions far more straightforward, bringing clarity to calculations involving powers.

Exponential Functions

Exponential functions are about repeated multiplication. They take the form \( a^b \) where \( a \) is the base and \( b \) is the exponent. The base \( e \) is special because it's natural. It appears in many natural growth processes, like population growth and radioactive decay.

• The expression \( e^{\pi} \) is such a function where \( e \) is the base.
• When you take the natural logarithm of an exponential function, like \( \ln e^{\pi} \), you're effectively undoing the exponentiation.
• The logarithm is the opposite operation to exponentiation, similar to how division is the opposite of multiplication.

The process of simplifying \( e^{\pi} \) with its logarithm leads us to use logarithmic properties to express complex exponential relationships in manageable terms.

Logarithmic Identities

Logarithmic identities are rules that simplify working with logarithms, helping to condense expressions into their neatest form. One cornerstone identity is \( \ln e = 1 \). This is because the logarithm asks, "To what power should the base \( e \) be raised to produce the number \( e \)?" The answer is 1.

• This identity is fundamental when working with natural logarithms.
• When simplifying \( \ln e^{\pi} \), recognizing that \( \ln e = 1 \) is key to getting to the final result of \( \pi \).
• It's crucial to remember this identity to efficiently solve problems and evaluate expressions in mathematics that involve natural logarithms.

Utilizing such logarithmic identities allows us to cut through complex expressions and see the underlying simplicity in mathematical operations.