Question

Determine whether the following statements are true and give an explanation or counterexample. Assume \(x>0\) and \(y>0\). a. \(\ln x y=\ln x+\ln y\). b. \(\ln 0=1\). c. \(\ln (x+y)=\ln x+\ln y\). d. \(2^{x}=e^{2 \ln x}\). e. The area under the curve \(y=1 / x\) and the \(x\) -axis on the interval \([1, e]\) is 1.

Short answer

Based on the analysis and solution provided, create a short answer question: Question: Determine if the following statement is true or false: \(\ln (x+y)=\ln x+\ln y\). Explain your answer.

Step-by-step solution

a. Determine if \(\ln x y=\ln x+\ln y\)

This statement is true because it is based on the property \(a \ln x=\ln x^{a}\), which is a property of logarithm. So using this property we have: \(\ln(x y) = \ln x + \ln y\)

b. Determine if \(\ln 0=1\)

This statement is false. The natural logarithm function is not defined for 0, as the natural logarithm is equivalent to finding the power to which e must be raised to obtain the given number. There is no power of e that will create a value of 0.

c. Determine if \(\ln (x+y)=\ln x+\ln y\)

This statement is false as well. A counterexample can be given by choosing two positive numbers \(x\) and \(y\). For example, let's take \(x=2\) and \(y=3\): \(\ln(2 + 3) = \ln5\) However, \(\ln 2 + \ln 3 = \ln(2 \cdot 3) = \ln 6\). Therefore, \(\ln 5 \neq \ln 6\)

d. Determine if \(2^{x}=e^{2 \ln x}\)

This statement is true. Using the property \(a \ln x=\ln x^{a}\), we can rewrite the equation as follows: \(2^{x}=e^{\ln x^{2}}\) Since \(2^x\) is just another way of writing \(x^2\), this equation becomes: \(x^2 = x^2\) Which is true.

e. Determine if the area under the curve \(y=1 / x\) and the \(x\) -axis on the interval \([1, e]\) is 1

This statement is true. In order to find the area under the curve within the given interval, we integrate the function \(y=\frac{1}{x}\) from \(1\) to \(e\). \(\int_1^e \frac{1}{x} dx\) The integral of \(\frac{1}{x}\) is \(\ln x\), so we evaluate the limit: \(\left[\ln x\right]_1^e = \ln e - \ln 1 = 1 - 0 = 1\) Hence, the area under the curve is equal to 1.

Key concepts

Logarithmic Identities

Understanding logarithmic identities is critical in mastering algebra and calculus. A fundamental identity is expressed as ewline ewline \(\ln(xy) = \ln(x) + \ln(y)\). This simply means that the natural logarithm of a product is equal to the sum of the natural logarithms of the individual factors.

Other crucial identities include the power rule, ewline ewline\(\ln(x^a) = a \cdot \ln(x)\), and the change of base formula, which is not solely for natural logarithms but applies to all logarithms, ewline ewline \(\log_b(x) = \frac{\ln(x)}{\ln(b)}\).

These identities provide a powerful toolkit for algebraic manipulation, allowing for the simplification of complex expressions, making them easier to work with or solve.

Defining Natural Logarithms

The natural logarithm, denoted as \(\ln(x)\), is a logarithmic function that uses \(e\) (Euler's number, approximately equal to 2.71828) as its base. In other words, \(\ln(x)\) answers the question: to what power must we raise \(e\) to get \(x\)?

It's important to note that the natural logarithm is only defined for positive real numbers, which explains why \(\ln(0)\) is undefined, as mentioned in step 2 of the solution. Concepts like continuity and limits are inherently tied to the properties of natural logarithms and are indispensable in calculus.

Counterexample in Mathematics

In mathematics, a counterexample is used to disprove a statement or a conjecture. By providing a single instance where the statement does not hold true, the entire statement is invalidated. In the context of logarithms, as shown in step 3 of the solution, the assumption that \(\ln(x+y) = \ln(x) + \ln(y)\) can be disproven by picking specific values for \(x\) and \(y\) and showing that the equality does not hold. A counterexample is a powerful tool in logical arguments, often used to emphasize the limitations of a rule or property.

Integration of 1/x

Integration is a fundamental concept in calculus, often referred to as finding the area under a curve. The function \(1/x\) is particularly significant because its integral is the natural logarithm of \(x\).

The definite integral of \(1/x\) from 1 to \(e\) represents the area under the curve of \(1/x\) within that interval. As we see in step 5 of the solution, the value of this integral is exactly 1, which corresponds to the fact that \(\ln(e)\) equals 1 since \(e\) is the base of the natural logarithms. Integrating functions like \(1/x\) provides a geometrical understanding of logarithmic functions and their applications.