Problem 95

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. $$ f(0)=0 $$

Step-by-Step Solution

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Answer

The statement is false because the natural logarithm of zero is undefined.

1Step 1: Interpret the Statement

The statement given is $ f(0)=0 $. This essentially means that if we substitute the value of $ x=0 $ into the function $ f(x)=\ln x $, we are supposed to get the output as 0.

2Step 2: Check the Validity of the Statement

However, a primary understanding of logarithmic functions is that they are not defined for the value 0. Meaning according to the logarithmic function $ \ln x $, there is no value of $ x $ for which $ f(x) $ would be 0.

3Step 3: Conclude the Statement

Therefore, the statement $ f(0)=0 $ is false, because $ f(0)=\ln(0) $ is undefined as logarithmic functions are not defined for 0.

Key Concepts

Natural LogarithmDomain of a FunctionUndefined Expressions

Natural Logarithm

The natural logarithm, denoted as $ \ln x $, is a logarithmic function with a base of $ e $, where $ e $ is an irrational constant approximately equal to 2.71828. The natural logarithm is unique due to its base and has numerous applications in mathematics, especially in calculus and exponential growth models. This function shows how exponentiation and logarithms are inversely related. Specifically, $ \ln x $ can help us find the exponent when $ e $ is raised to a power to get $ x $. For instance, if $ e^y = x $, then $ y = \ln x $. Some important properties of natural logarithms include:

$ \ln(1) = 0 $ because $ e^0 = 1 $
$ \ln(e) = 1 $ because $ e^1 = e $
It is only defined for positive real numbers ($ x > 0 $)

Throughout mathematics, $ \ln x $ is crucial for solving problems involving continuous growth and decay since its derivative and integral are both elegantly simple compared to other logarithmic bases.

Domain of a Function

Understanding the domain of a function is essential for correctly solving mathematical problems. The domain refers to all the permissible input values that a function can accept. For the natural logarithm function, $ \ln x $, the domain only includes positive real numbers: $ x > 0 $. This means $ \ln x $ cannot take zero or negative numbers as inputs. If you try to input $ x = 0 \, \text{or} \, x < 0 $, the result is undefined. This restriction arises because the logarithm measures how many times we multiply the base, $ e $, to get to $ x $. Since zero and negative numbers do not have meaningful values in this context, they fall outside the domain. When dealing with any function, clearly identify its domain as a first step:

Check for restrictions like division by zero.
Ensure all inputs yield real, defined values.

Paying attention to the domain is particularly critical with $ \ln x $, preventing errors when analyzing or graphing the function.

Undefined Expressions

Undefined expressions, such as $ \ln 0 $, occur when a function receives an input outside its domain. In our exercise, the expression $ f(0) = \ln(0) $ is undefined because $ \ln x $ cannot take zero as an argument according to its domain rules. Here are some points to consider when a function yields undefined expressions:

Always check the domain of the function first to avoid using invalid inputs.
If the calculation results in a division by zero or a logarithm of zero/negative number, it will be undefined.
Understanding why expressions are undefined can help prevent incorrect assumptions in problem-solving, such as assuming $ f(0) = 0 $ when dealing with $ \ln x $.

Being cautious and evaluating functions within their domain helps ensure valid results and a correct mathematical understanding.

Problem 94

Problem 96

Other exercises in this chapter

Problem 93

In Exercises, use a graphing utility to verify that the functions are equivalent for $x>0$. $$ \begin{aligned} &f(x)=\ln \frac{x^{2}}{4} \\ &g(x)=2 \ln x-\ln

View solution

Problem 94

In Exercises, use a graphing utility to verify that the functions are equivalent for $x>0$. $$ \begin{aligned} &f(x)=\ln \sqrt{x\left(x^{2}+1\right)} \\ &g(x)

View solution

Problem 96

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

View solution

Problem 97

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

View solution