Problem 80 - Thomas Calculus

Question

The inverse relation between \(e^{x}\) and \(\ln x\) Find out how good your calculator is at evaluating the composites $$ e^{\ln x} \text { and } \ln \left(e^{x}\right) $$

Step-by-Step Solution

Answer

Both \(e^{\ln x}\) and \(\ln(e^x)\) yield back the input value, confirming precise inverse operations.

Step 1: Understand the Functions

The function \(e^x\) is the exponential function with base \(e\), and \(\ln x\) is the natural logarithm function. These two functions are inverses of each other, which means that applying one after the other should return the original input, provided it lies within the domain of the functions.

Step 2: Analyze the Composite \(e^{\ln x}\)

For the function \(e^{\ln x}\) where \(x > 0\), you first apply \(\ln x\), which converts \(x\) back to an exponent of \(e\). Then, \(e^{\ln x} = x\) because the \(e\) and \(\ln\) functions cancel each other out being inverse operations, as long as \(x > 0\).

Step 3: Analyze the Composite \(\ln(e^x)\)

For the function \(\ln(e^x)\), you first calculate \(e^x\), which gives you a number in the form \(y=e^x\). Applying \(\ln\) converts this back to \(x\) since \(\ln(e^x) = x\). Again, this is true because they are inverse functions working over the domain of all real numbers \(x\).

Step 4: Confirm with a Calculator

Use a calculator to compute \(e^{\ln 10}\) and \(\ln(e^2)\). For \(e^{\ln 10}\), input \(\ln 10\) first, find its value, and then apply the exponential function. You should get approximately 10. Similarly, for \(\ln(e^2)\), first calculate \(e^2\) and then find the natural logarithm, which should be about 2.

Key Concepts

Exponential Function

The exponential function, denoted as \(e^x\), is a mathematical function where \(e\) is a constant approximately equal to 2.71828. This constant is unique because it is the base of natural logarithms, meaning that the growth rate of the function is proportional to its value at any point. In simple terms, if you change the input \(x\) slightly, the output will change proportionately, a property widely used in natural sciences and growth models.
Some points to note about the exponential function are:

• The domain is all real numbers, while the range is \((0, \infty)\).
• It rapidly increases as \(x\) becomes large, and approaches zero as \(x\) becomes negative.
• It is continuous and differentiable across its domain.

The function forms the basis for logarithmic calculations, acting as the inverse to natural logarithms, enabling back-and-forth transitions between exponent and natural log forms.

Natural Logarithm

The natural logarithm, represented as \(\ln x\), is defined as the logarithm to the base \(e\). It is the inverse of the exponential function, meaning \(\ln(e^x) = x\) and \(e^{\ln x} = x\) if the domains are properly respected, ( where the exponential by its nature can take any real number, and the natural log requires \(x > 0\). Some features of the natural logarithm include:

• Its domain is \((0, \infty)\), covering all positive real numbers.
• The range extends over all real numbers because it maps very small inputs to large negative numbers and large inputs to large positive numbers.
• Like the exponential function, it is continuous and differentiable.

Because of its property of transforming multiplicative processes into additive ones, it is invaluable in fields like mathematics, physics, and engineering for simplifying complex multiplicative relationships.

Composite Functions

Composite functions result from the combination of two functions where the output of one function becomes the input to another. For example, if \(f(x)\) and \(g(x)\) are two functions, their composite is expressed as \(f(g(x))\).
When dealing with inverse functions like \(e^x\) and \(\ln x\), their composites are especially interesting:

• \(e^{\ln x}\): Start with any positive number \(x\), apply the natural log to bring it into the exponent form, and then revert it back using the exponential function. The result is the original number \(x\) because the two functions cancel each other out.
• \(\ln(e^x)\): Begin with a real number \(x\), calculate its exponential form \(e^x\), and apply the natural log to bring it back to \(x\). Again, they nullify each other due to their inverse properties.

The beauty of composite functions, especially with inverse pairs like these, lies in their ability to simplify complex calculations, making them easy to handle if the properties and domains are well understood.