Problem 22 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(\ln e^{8}\)

Step-by-Step Solution

Answer

The evaluated expression is 8.

Step 1: Understand the Problem

We need to evaluate the expression \( \ln e^{8} \). Recall that \( \ln \) denotes the natural logarithm, which is the logarithm with base \( e \).

Step 2: Apply the Logarithm Rule

One important logarithm property to remember is that \( \ln e^{x} = x \). This is because the logarithm of a number raised to a power is the power itself, when the base of the logarithm is the same as the base of the exponent.

Step 3: Simplify the Expression

Using the property from Step 2, we apply it directly to the expression: \( \ln e^{8} = 8 \). This is because the exponent on \( e \), which is 8, becomes the result of the natural logarithm.

Key Concepts

Logarithm Rules

Logarithms are an essential concept in mathematics that help us deal with large numbers more easily. In the context of natural logarithms, the base is the irrational number \( e \), approximately equal to 2.71828. Knowing some fundamental logarithm rules can make simplifying logarithmic expressions much easier.

Here are key rules to remember:

• Logarithm of a Product: \( \ln(ab) = \ln a + \ln b \)
• Logarithm of a Quotient: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \)
• Logarithm of a Power: \( \ln(a^b) = b \ln a \)
• Logarithm of \( e \): \( \ln e = 1 \)
• Logarithm of \( e^x \): \( \ln e^x = x \)

The last rule is particularly useful in the problem we are considering because when the base of the exponential function is the same as the base of the logarithm, they effectively cancel each other out, leaving just the exponent.

Power Rule

The power rule is a fundamental principle that is not only crucial when working with logarithmic and exponential functions but also with calculus. The general rule is if you have a power inside the logarithm, you can bring that power in front of the logarithm for easier computation.

For natural logarithms, it looks like this:

• \( \ln(a^b) = b \ln a \)

This rule is useful because it allows you to transform a potentially complex expression into a linear term, simplifying algebraic manipulations. In the expression \( \ln e^8 \), the entire evaluation hinges on understanding that \( e \) and \( \ln \) interact in a way dictated by the power rule applied specifically for \( e^x \), namely that \( \ln e^{x} = x \).

Understanding this rule can dramatically simplify solving logarithms, especially when the base of the logarithm matches the base of the exponent.

Exponential Functions

Exponential functions are powerful mathematical tools that describe many phenomena of growth and decay, like population growth or radioactive decay. An exponential function has the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm.

The number \( e \), Euler's number, is important in mathematics because it is the unique rate where the growth process remains proportional over time. Exponential functions are special because they have properties that other functions don't, such as a constant rate of growth represented by their derivatives.

In the expression \( \ln e^8 \), we are dealing with an exponential function \( e^8 \). The concept of an exponential function with base \( e \) allows the natural logarithm \( \ln \) to interact with it simply and directly, reducing the complexity of the expression by directly yielding the exponent, 8 in this case. When learning about exponential functions, it is essential to see how they harmonize with natural logarithms to simplify and solve expressions easily.