# What is the integral 1/x log(x) or lnx?

• pb23me
In summary, the conversation discusses the confusion surrounding the use of logarithms in Wolfram Alpha. It is explained that in Wolfram, log(x) refers to the natural logarithm, while other bases may be denoted as LOG. The derivative of ln(x) is also mentioned, and it is suggested to pay attention to the notation used in the equation box. It is also noted that the software Mathematica, developed by Wolfram, uses LOG instead of the more commonly used LN for the natural logarithm.

## Homework Statement

I know it is supposed to be lnx however I find something peculiar. When I integrate it in wolfram alpha they give the integral as log(x). What the heck is going on here?

## The Attempt at a Solution

If you check closer on Wolfram, by log(x) they mean the natural logarithm.

ok thanks, When looking at more complex solutions sometimes there are many logx's how am I supposed to know which one is logx and which one is not really logx?

They will usually define what they use. As a rule of thumb, I find that they always tend to use the natural log, but it could happen that they don't. Just pay attention to the bottom of the equation box thingy.

the derivative of ln(x) is "one over the thing inside, times the derivative of the thing inside" -- my calc professor

so, y = ln (x); y' = 1/x * 1 = 1/x

i'm not sure if this helps or not... you are asking for y given y' = 1/x ?

ok thanks guyz I precciate ya!

It's simple. The 'Mathematica' software developed by Wolfram himself or his company has the natural logarithm (aka neperian logarithm) denoted as LOG, instead of the widely used LN which is derived from <natural logarithm> spelled in Latin. Most people use LOG for the logarithm in other base than Euler's number 'e'.

## 1. What is the definition of the integral 1/x log(x) or lnx?

The integral of 1/x log(x) or lnx is the antiderivative of the function 1/x log(x) or lnx, which is the function that, when differentiated, gives the original function as its result. In other words, it is the inverse operation of differentiation.

## 2. Can the integral 1/x log(x) or lnx be solved analytically?

Yes, the integral 1/x log(x) or lnx can be solved analytically using integration by parts. This method involves breaking down the integral into simpler parts and using a formula to solve each part. However, the resulting integral may still be difficult to evaluate without the use of numerical methods.

## 3. How is the integral 1/x log(x) or lnx used in real-world applications?

The integral 1/x log(x) or lnx is used in various fields of science and engineering, such as physics, chemistry, and economics. It is used to calculate quantities such as work, energy, and population growth, which are essential in understanding natural phenomena and making predictions.

## 4. What is the domain and range of the function 1/x log(x) or lnx?

The domain of the function 1/x log(x) or lnx is all positive real numbers greater than zero, since the natural logarithm of a negative number is undefined. The range of the function is all real numbers, as the natural logarithm can take on any real value.

## 5. Are there any special properties or rules to keep in mind when working with the integral 1/x log(x) or lnx?

One important property to keep in mind is that the integral 1/x log(x) or lnx is not defined at x = 0, as the function 1/x becomes undefined at this point. Another property is that the integral is considered an improper integral, meaning that it does not converge to a finite value and may require special techniques to evaluate.