VIDEO: e ^ ln (x) = x

The natural logarithm ln (x)

In high school mathematics, the exponential function is often f (x) = e^x, which is based on Euler's number e (about 2.71). Historically, this unusual number can be explained as the result of a compound interest problem.

• There is an inverse function for this exponential function, namely the natural logarithm f (x) = ln x (you can put the variable "x" in brackets here, but you don't have to).
• The following rule of thumb is easy to understand: The exponential function forms Potencies, the logarithm function "asks" for the exponent.

But why is e ^ ln (x) = x?

The expression "e ^ ln (x) = x" looks like it should scare people with little mathematical training. This is not the case, however, because the expression is easy to understand:

• First of all, it should be rewritten as e ^ ln (x) = e^{ln x} = x. In other words: if you take the inverse function of e^x, namely ln x to the power of the exponential function, the variable "x" comes out again.


Reverse the logarithm - that's how it works

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The inverse function of the logarithm is not difficult to determine. You have to ...

• The reason is that function and inverse function cancel each other out. (Root (x)) ² = x, because the root function and the square function cancel each other out.
• The equation is a bit astonishing, however. In addition to this more understandable justification, the correctness of the equation can also be proven that e ^ ln (x) = x. To do this, form the natural logarithm on both sides of the equation and get ln (e^{ln x}) = ln x. On the left side you apply the well-known logarithmic laws: ln x _* lne = lnx (since ln e = 1).
• The opposite conclusion is also interesting. Namely, "ln (e^x) = x ", which can be shown by direct application of the logarithmic laws.

But where do such mathematical expressions occur or are they needed?

• The simpler expression "ln (e^x) = x "is required if you Exponential equations want to resolve (you can get to the exponent you are looking for by taking the logarithm).
• The more complicated expression e^{ln x} = x is required when one Equations should solve, for which the required quantity x is in the logarithm (here one comes by exponentiation, i.e. by applying the exponential function to the unknown x).