Problem 33 - Contemporary Precalculus

Question

Evaluate the given expression without using a calculator. $$\ln e^{x+y}$$

Step-by-Step Solution

Answer

Question: Evaluate the expression $$\ln e^{x+y}$$. Answer: The expression $$\ln e^{x+y}$$ simplifies to $$x+y$$.

Step 1: Identify the properties of logarithms and exponential functions

To evaluate the given expression, we will use the relationship between logarithmic and exponential functions which states, for any number 'a' and base 'b': $$\log_b b^{a} = a$$ In this case, the base of the logarithm function is "e".

Step 2: Apply the relationship to the given expression

Now we will apply the relationship mentioned in Step 1 to the given expression: $$\ln e^{x+y}$$ Here, the base of the logarithm is "e" and the exponent is \((x+y)\). According to the relationship, this simplifies to: $$\ln e^{x+y} = x+y$$

Step 3: Present the final solution

After simplifying the given expression, we arrive at the final solution: $$\ln e^{x+y} = x+y$$

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us solve equations where the unknown variable is the exponent. When we talk about the logarithm, we usually mean logarithm to the base 10 unless otherwise specified, but in mathematics, the natural logarithm (base \(e\)) is often used.

• The logarithmic function \(\log_b(x)\) tells us the power to which the base \(b\) must be raised to obtain \(x\).
• This means that \(\log_b(b^a) = a\), i.e., the logarithm of a number raised to the base is simply the exponent.
• A very common property is that \(\log(a \cdot b) = \log(a) + \log(b)\), which can be useful in simplifying expressions.

Logarithmic functions have important applications in various fields such as science, engineering, and financial analysis due to their ability to handle large numbers and exponential growth. Understanding how logarithms work is crucial for handling exponential equations effectively.

Exponential Functions

Exponential functions describe growth or decay processes in which the rate of change is proportional to the current value. The form of an exponential function is \(f(x) = a \cdot b^x\), where \(a\) is a constant and \(b\) is the base of the exponential.

• If \(b > 1\), the function represents exponential growth.
• If \(0 < b < 1\), the function shows exponential decay.
• A continuous growth or decay process is often modeled by the base \(e\), making the function \(f(x) = e^{x}\) significant in natural contexts.

Exponential functions are pervasive in the real world, modeling phenomena from population growth to radioactive decay. Understanding how these functions behave enables us to predict and analyze real-life situations effectively.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a logarithm with an irrational base \(e\) (approximately 2.718). It's an essential concept in calculus and mathematical analysis, vital for its simpler properties when involved with integration and differentiation.

• One key property is that \(\ln(e^a) = a\), which helps in simplifying expressions involving powers of \(e\).
• Natural logarithms are used broadly in compound interest problems, population dynamics, and even thermodynamics.
• Perhaps their greatest utility lies in transforming exponential growth into linear growth, making data analysis simpler.

Applications of natural logarithms are vast, spanning various disciplines such as economics, physics, and ecology. Understanding them allows for effective handling of continuously growing or decaying systems.