Logarithms are known as the inverse operations of exponential functions of the same base, so the two types of functions undo each other. It does not matter whether you apply the logarithm or the exponent first—the result is the same. If the base of the logarithm is Euler’s number, \(e\), there are special properties that the function has. It is called the natural logarithm, and uses the notation \(\ln\) to reflect upon that. To demonstrate the base of the natural logarithm:
$$\ln (e) = 1$$ $$\ln(e^a) = a$$
Natural logarithms follow all the properties that other logarithms do, but there are some special patterns that can be observed. This article will demonstrate the standard properties of logarithms with the natural logarithm, and then proceed to show properties exclusively for the natural logarithm.
As stated before, all the identities that apply to other logarithms apply to the natural logarithm. This set of examples will show those identities being applied to expressions with natural logarithms:
Example 1: \(\ln(4) + \ln(8) = \ln(4 \cdot 8) = \ln(32)\)
Example 2: \(\ln(36) = \ln(\frac{72}{2}) = \ln(72) - \ln(2)\)
This last example is especially noteworthy, because it shows how you can eliminate a logarithm with an undesirable base:
Example 3: \(\ln(4) \cdot \log_{4}(e^2) = \log_{e}(4) \cdot \log_{4}(e^2) = \ln(e^2) = 2\)
The only catch is that you need to make the base of the uncommon logarithm match the argument of the natural logarithm through a combination of logarithmic identities, unless the expression is already given in that form.
The natural logarithm also has a unique derivative:
$$\frac{d}{dx}(\ln (x)) = \frac{1}{x}$$
However, since the natural logarithm is only defined for positive numbers, when comparing these two functions, we set a restriction on \(\frac{1}{x}\) of \(x > 0\).
Example 4: Find the derivative of \(\ln(2x + 1)\).
Solution: Use the Chain Rule and our identity above to get
$$\frac{d}{dx}(\ln(2x + 1)) = \frac{2}{2x + 1}$$
This derivative also indicates a special property of the natural logarithm: it is monotonically increasing, or constantly increasing. This is because the derivative is positive on the natural logarithm’s entire domain, so the natural logarithm will always be increasing.
A Taylor Series is a polynomial with an infinite number of terms equivalent of a function that is not a polynomial, constructed using derivatives of the function (as in first derivative, second derivative, etc.). The natural logarithm requires a special Maclaurin Series (a Taylor Series that gets centralized at \(a = 0\)), because \(\ln(0)\) is undefined:
$$\ln (1 - x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - …$$
Example 5: Consider the Taylor Series shown above. Differentiating that entire equation gives
$$\frac{-1}{1 – x} = -1 - x – x^2 – x^3 - … $$
Now divide this by \(-1\) to get
$$\frac{1}{1 – x} = 1 + x + x^2 + x^3 + …$$
This is interesting because this is a very common Taylor Series itself.
Example 6: In the Maclaurin Series for \(\ln(1 - x)\), substitute \(x\) for \(-x\) to get another Taylor Series:
$$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - …$$
Some Calculus II teachers believe that these three Taylor Series are worth memorizing; they are common enough to make it worthwhile.
You have seen a variety of examples and demonstrations of the unique traits of the natural logarithm function, ranging from algebra to calculus. Still, it will be difficult to understand the roots of these topics if you forget that \(\ln(e) = 1\).
A reference, courtesy of
"The Art of Problem Solving Intermediate Algebra" Rusczyk, Richard and Crawford, Matthew