Question

In Exercises, determine whether the statement is true or false given that \(f(x)=\ln x .\) If it is false, explain why or give an example that shows it is false. $$ \text { If } f(x)<0, \text { then } 0

Short answer

The statement 'If \( f(x) < 0 \), then \( 0 < x < 1 \)' is true.

Step-by-step solution

Understand the Properties of Natural Logarithm

The natural logarithm function,\( \ln x \), is only defined for x > 0. It is a decreasing function in the interval (0, 1), meaning the values of \( \ln x \) are less than 0 when \( 0 < x < 1 \) and more than 0 when x > 1.

Analyze the Given Statement

The given statement says: if \( f(x) < 0 \), then \( 0 < x < 1 \). This translates to: if \( \ln x < 0 \), then \( 0 < x < 1 \). We want to determine if this statement is true or false.

Validate the Statement

Recall from our understanding of natural logarithm properties that the statement is indeed correct. For \( \ln x \) to be less than 0, \( x \) indeed must fall within the interval (0, 1). Thus, the statement is true.

Key concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have a variety of applications in both pure and applied mathematics. The natural logarithmic function, denoted as \(\text{ln} x\), is particularly important because it is the inverse of the exponential function \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828.

This special logarithm has a base \(e\) and is used to determine the time needed for an investment to grow to a certain amount, to solve problems involving population growth, and even to model the spread of diseases. Understanding the behavior of \(\ln x\), such as the fact that it is undefined for \(x\leq 0\), increases without bound as \(x\) approaches infinity, and decreases without bound as \(x\) approaches zero from the right, is pivotal in grasping not just the function itself, but how we model real-world phenomena using mathematics.

Importantly for calculus students, the derivative of \(\ln x\) with respect to \(x\) is \(1/x\), and this forms the basis for many techniques in integration involving logarithmic functions.

Inequalities in Calculus

Inequalities play a crucial role in calculus, particularly when defining intervals of increase or decrease and when working with optimization problems. When we talk about inequalities in calculus, we are often concerned with finding the range of values for which a function takes on certain characteristics. For instance, understanding when a function is less than or greater than a particular value can help us delineate regions of interest on a graph.

More formally, if we have a continuous function \(f(x)\), we can analyze inequalities such as \(f(x) > a\) or \(f(x) < b\) to determine the intervals where the function lies above a horizontal line \(y = a\) or below a horizontal line \(y = b\), respectively. This becomes especially useful in conjunction with the first and second derivatives, which tell us about the slope of the function's tangent line and the function's concavity, thereby allowing to predict and describe its behavior with greater precision.

Properties of Natural Logarithms

The properties of natural logarithms are fundamental to understanding how these functions behave and interact with other mathematical operations. One key property, as showcased by our exercise, is the relationship between the natural logarithm and its input values.

The natural logarithm of a number less than 1 is negative, and the natural logarithm of a number greater than 1 is positive. This is because the function \(\ln x\) roughly measures the amount of time it takes for the quantity \(e^t\) to grow to \(x\), and negative time indicates a growth that occurred before the start (which corresponds to the time at \(e^0\), or 1).

Other key properties of natural logarithms include:

• \(\ln(xy) = \ln(x) + \ln(y)\) - shows the logarithm of a product as the sum of logarithms.
• \(\ln(x^y) = y\ln(x)\) - allows us to remove the exponent of a logarithm argument.
• \(\ln(1) = 0\) - states that the natural logarithm of 1 is always zero because any number to the power of zero equals one.
• \(\ln(e) = 1\) - because the natural logarithm is the inverse of \(e^x\), the natural logarithm of \(e\) is one.

These properties are not just abstract rules but tools that help solve a wide array of problems in mathematics, from simple algebraic equations to complex calculus integrals.