Problem 24 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \ln e^{5} $$

Step-by-Step Solution

Answer

The exact value of \( \ln e^5 \) is 5.

Step 1: Understand the Logarithmic Property

One of the fundamental properties of logarithms is that the natural logarithm of an exponential function with the same base simplifies as follows:\[\ln(e^x) = x\] This is because the natural logarithm, \(\ln\), is the inverse function of the exponential function base \(e\).

Step 2: Apply the Logarithm Property

Apply the logarithmic property explained in Step 1 directly to the given expression:\[\ln(e^5) = 5\] Thus, the logarithm simplifies to the exponent itself, which is 5 in this case.

Key Concepts

Exponential Function

Exponential functions are a crucial concept in mathematics. They are functions that grow at an exponential rate, meaning the quantity increases rapidly. An exponential function has the general form \( f(x) = a \, e^{bx} \), where \( a \) is a constant, \( b \) is the base of the exponent, and \( e \) is the Euler's number, approximately equal to 2.71828.
The base \( e \) is used since it has unique properties that simplify mathematical calculations, especially in calculus.

• Growth and decay are easily modeled with exponential functions.
• The function is represented in the form \( f(x) = e^x \).

These functions are continuous and smooth, providing a natural model for many real-world processes such as population growth and radioactive decay. Understanding how exponential functions operate can aid in grasping their inverse, the logarithm.

Logarithmic Properties

Logarithmic properties are rules that apply to logarithms, helping simplify complex expressions. One of the most used properties is \( \ln(e^x) = x \), which shows how logarithms and exponentials are related. This rule stems from the relationship between exponential functions and logarithms, as they are inverses.

• Logarithms turn multiplication into addition: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
• They transform division into subtraction: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• They convert exponentiation into multiplication: \( \ln(a^b) = b \cdot \ln(a) \).

These properties are invaluable in solving logarithmic equations and simplifying expressions. In the given exercise, understanding these properties makes it clear why \( \ln(e^5) = 5 \) simplifies so easily.

Inverse Functions

Inverse functions reverse the operation of the original function. When a function \( f \) maps \( x \) to \( y \), its inverse \( f^{-1} \) will map \( y \) back to \( x \). For example, the inverse of an exponential function is the logarithm, and vice versa.

• The exponential function \( e^x \) and the natural logarithm \( \ln(x) \) are inverse functions.
• The property \( \ln(e^x) = x \) illustrates this inverse relationship.
• Applying a function and its inverse results in the original input value, e.g., \( \ln(e^5) = 5 \).

Inverse functions are essential in various fields, allowing for the solving of equations where it is necessary to reverse mathematical operations. They provide clarity and exactness, especially in processes involving growth and decay, trigonometry, and more.