Problem 44 - Elementary and Intermediate Algebra

Question

Evaluate each expression without using a calculator. $$ \ln e^{4} $$

Step-by-Step Solution

Answer

The value of \( \ln e^4 \) is 4.

Step 1: Understanding the Problem

The expression given is \( \ln e^{4} \). We need to evaluate this expression without using a calculator.

Step 2: Applying the Logarithmic Identity

Recall the logarithmic identity: \( \ln(a^b) = b \cdot \ln(a) \). In this case, we apply \( \ln(e^4) = 4 \cdot \ln(e) \).

Step 3: Using the Natural Logarithm Property

The natural logarithm \( \ln(e) \) is equal to 1 because the natural logarithm is the inverse of the exponential function with base \( e \). Therefore, \( \ln(e) = 1 \).

Step 4: Substitute and Simplify

Now substitute \( \ln(e) = 1 \) into our earlier expression: \( 4 \cdot \ln(e) = 4 \cdot 1 = 4 \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special logarithm that has the base of the mathematical constant \( e \). This constant \( e \) is approximately equal to 2.71828 and arises naturally in various mathematical contexts, such as in calculations of compound interest, population growth models, and complex numbers. The natural logarithm of a number \( x \), \( \ln(x) \), gives us the power to which \( e \) must be raised to equal \( x \).

For example:

• \( \ln(1) = 0 \) because \( e^0 = 1 \)
• \( \ln(e) = 1 \) because \( e^1 = e \)

The natural logarithm is particularly useful in solving problems where exponential growth or decay is involved as it simplifies many expressions involving powers of \( e \). Understanding the natural logarithm and its properties allows you to interpret equations and simplify them effectively, as seen in the exercise \( \ln(e^4) \).

Learning about natural logarithms will give you insights into a wide variety of scientific and engineering applications.

Logarithmic Identity

Logarithmic identities simplify complex expressions and provide useful properties for manipulation. One of the key identities is the power rule: \( \ln(a^b) = b \cdot \ln(a) \). This identity helps in breaking down exponential terms and makes calculations easier and more intuitive.

Because logarithms have properties similar to exponents, these identities allow you to transform multiplicative situations into additive ones, which are generally simpler to handle. For example:

• If you have \( \ln(x^3) \), it becomes \( 3 \cdot \ln(x) \).
• The expression \( \ln(e^4) \) becomes \( 4 \cdot \ln(e) \), which we already know is 4 because \( \ln(e) = 1 \).

These transformations make solving logarithmic equations a systematic task, turning complex calculations into manageable steps. Knowing how to use such identities effectively will reduce computation complexity and show the relationships within exponential functions clearly.

Exponential Function

An exponential function is a mathematical expression of the form \( f(x) = a^x \) where \( a \), the base, is a positive real number. The function \( f(x) = e^x \) uses the special constant \( e \) as its base, making it the natural exponential function.

The exponential function is prominent because it describes continuous growth and decay processes like population growth, radioactive decay, and interest calculations. In mathematical expressions or equations, the exponential function allows us to model real-world scenarios where growth rates are proportional to the current value.

For instance, the property that makes the exponential function \( e^x \) uniquely powerful is its derivative, which is itself \( e^x \). This property results in simplification when solving differential equations and explains why the natural logarithm \( \ln(x) \) serves as the inverse to \( e^x \).

A key takeaway is the reciprocal nature between the exponential and logarithmic functions, especially using \( e \), which is instrumental in both simplifying expressions and solving equations involving exponential terms.