Question

Give a geometric interpretation of the function $\ln x={\int }_1^x\frac{dt}{t}$.

Short answer

Answer: The geometric interpretation of the function $\ln x={\int }_1^x\frac{dt}{t}$ is that the area under the curve of the function $y=\frac{1}{t}$ from 1 to x on the horizontal axis is equal to the natural logarithm of x.

Step-by-step solution

Define the function

The function we are working with is $\ln x={\int }_1^x\frac{dt}{t}$. The natural logarithm function, denoted by $\ln x$, is the inverse of the exponential function $e^x$ and its integral from 1 to x of the function $\frac{1}{t}$ provides a geometrical interpretation.

Area under the curve interpretation

To interpret this, let's consider that the integration represents the area under the curve. So, the area under the curve of the function $y=\frac{1}{t}$ from 1 to x is equal to the natural logarithm of x.

Sketch the curve

To get a better glimpse of what we're analyzing, sketch the curve $y=\frac{1}{t}$ for $t\ge 1$. The curve starts at $t=1$, $y=1$ and decreases as the value of t increases. It forms a hyperbolic shape and never touches the horizontal axis.

Shading the region

Now, shade the area under the curve starting from $t=1$ until some arbitrary value of $t=x$ on the horizontal axis. The shaded region will be bounded by the curve $y=\frac{1}{t}$, the horizontal axis, and the vertical lines $t=1$ and $t=x$.

Geometric interpretation of the integral

The area of the shaded region is equal to the integral of the function from 1 to x, which is ${\int }_1^x\frac{dt}{t}$. By the relation given in the exercise, this integral also represents the natural logarithm of x, or $\ln x$. Therefore, the area of the shaded region has a geometric interpretation of the natural logarithm function. In summary, the geometric interpretation of the function $\ln x={\int }_1^x\frac{dt}{t}$ is that the area under the curve of the function $y=\frac{1}{t}$ from 1 to x on the horizontal axis is equal to the natural logarithm of x.

Key concepts

Natural Logarithm

The natural logarithm, often denoted as $\ln x$, is a special function in mathematics. It is the logarithm to the base of the mathematical constant $e$, where $e$ is approximately equal to 2.71828. Unlike the common logarithm which uses the base 10, the natural logarithm is used widely across calculus and higher mathematics because of its unique properties.
The natural logarithm is basically the inverse operation to exponentiation. In other words, if you have an exponential function like $e^y=x$, then $y=\ln x$. This means that taking the natural logarithm of a number $x$ corresponds to finding what power you must raise $e$ to get $x$.

• Natural Logarithms appear frequently in growth and decay problems in science and engineering.
• They are essential in solving equations involving exponential functions.

Understanding the natural logarithm helps to realize its connection with the growth rate of functions and its association with the integration of reciprocals.

Integration

Integration is a fundamental concept in calculus that involves finding the "total" or the "sum" over a specific interval.
In simple terms, integration can be thought of as a process of finding the area under a curve. For the exercise, integration is used to find the area under the curve of the function $y=\frac{1}{t}$ from 1 to $x$. The mathematical notation for this process is ${\int }_1^x\frac{dt}{t}$.
Integration has several applications in real life:

• Calculating areas and volumes
• Solving differential equations
• Analyzing motion and change

The integral, as applied to the function $\frac{1}{t}$, not only provides the area but also directly corresponds to the natural logarithm of $x$, illustrating the powerful relationship between these mathematical concepts.

Area Under the Curve

The concept of area under the curve is a key part of understanding the integral ${\int }_1^x\frac{dt}{t}$ and its relationship with the natural logarithm function.
Imagine graphing the function $y=\frac{1}{t}$ starting from $t=1$. The curve forms a downward-sloping hyperbola, and the "area under the curve" is bounded by this hyperbola, the horizontal axis, and the vertical lines at $t=1$ and $t=x$.
Shading this region helps visualize what the integral calculates:

• It is the total area enclosed by these boundaries.
• It represents the accumulated "sum" of the function $\frac{1}{t}$ from 1 to $x$.

This shaded area under the curve is exactly what the natural logarithm function measures for any value of $x$ greater than 1, effectively demonstrating the close relationship between integration and logarithms through geometric interpretation.

Inverse Function

An inverse function essentially reverses another function's operation. For instance, if you have a function $f$ that transforms $x$ to $y=f(x)$, then its inverse $f^{-1}$ would convert the $y$ back to $x$.
In the context of the natural logarithm, the function $e^x$ is its inverse. If $e^x$ gives you outputs on an exponential curve, $\ln x$ determines the input $x$ that would raise $e$ to a particular height $y$.

• This relationship highlights how exponentials and logarithms are deeply intertwined.
• Knowing this inverse relationship helps solve equations involving exponential growth or decay.

Understanding inverse functions is crucial for recognizing how complex numbers and transformations connect, allowing for simpler calculations in a variety of mathematical and scientific contexts.