Problem 24 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\ln e^{x-z}$$

Step-by-Step Solution

Answer

The result of evaluating the expression \(\ln e^{x-z}\) without a calculator is \(x - z\).

Step 1: Understanding the properties of logarithms and exponentials

Firstly, it is important to note that \(ln(e^a) = a\). This is because the natural logarithm, signified by \(ln\), is the inverse function to the exponential function \(e^a\). This means when you apply a natural logarithm to an exponential that has the same base \(e\), we just get \(a\) back out. This will be applied directly in the next step.

Step 2: Applying properties of logarithms

We substitute the expression \(e^{x-z}\) in to \(a\) from step 1 to obtain: \[ \ln e^{x-z} = x - z \] Hence the resulting expression after applying the property of logarithms is just \(x-z\).

Step 3: Writing down the final result

From step 2, after applying the property of exponentials and logarithms, the final result is \(x - z\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithmic function that specifically uses \( e \) (Euler's number) as its base. Euler's number \( e \) is approximately equal to 2.71828, and it is an irrational number frequently used in mathematical calculations because of its unique properties.

The natural logarithm has a unique and interesting property: it effectively "undoes" the effect of its corresponding exponential expression. This is because the natural logarithm and the exponential function with base \( e \) are inverses of each other.

If we have a natural log of an expression involving \( e \), such as \( \ln(e^a) \), the result simplifies directly to just \( a \). Here, no calculation is involved other than recognizing the inverse property.

• The natural logarithm \( \ln \) returns the "power" to which the base \( e \) must be raised to get the original number.
• For example, \( \ln(e^3) = 3 \) because \( e^3 \) raised to the power of the natural logarithm simplifies to 3.

Exponential Function

The exponential function is a mathematical function of the form \( e^x \), where \( e \) is Euler’s number and \( x \) is any real number. This function is widely used in various fields, such as biology, economics, and physics, due to its ability to model growth and decay processes efficiently.

Key features of the exponential function include:

• It grows (or decays) at a rate proportional to its current value, which makes it an essential tool for modeling exponential growth or decay.
• The function is always positive, meaning \( e^x > 0 \) for any real number \( x \), and it is never zero.
• The graph of the exponential function is a smooth, continuous curve that becomes steeper and steeper as \( x \) increases.
• When \( x = 0 \), \( e^x = 1 \) since any number raised to the power of 0 is 1.

The exponential function plays a critical role in calculus, as its derivative with respect to \( x \) is itself \( e^x \), which is a distinctive property not shared by other functions.

Inverse Functions

Inverse functions are pairs of functions that "undo" each other. For any given function \( f(x) \), if \( g(x) \) is its inverse, then applying both functions in sequence will return the original value: \( f(g(x)) = x \) and \( g(f(x)) = x \).

In our context of natural logarithms and exponential functions, we see this relationship clearly:

• The natural logarithm \( \ln(x) \) and the exponential function \( e^x \) are known as inverse functions because they reverse each other's operations.
• This means if you take the natural logarithm of an exponential function, \( \ln(e^a) \), it will return \( a \). Conversely, raising \( e \) to the power of the natural logarithm of \( x \), \( e^{\ln(x)} \), returns \( x \).

Understanding inverse functions is crucial because it allows us to solve complex equations by isolating the variable of interest. This inverse relationship simplifies computations and is fundamental in algebra and calculus.