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13.4 | Natural Logarithm Function and Its Graph |

In these revision notes for Natural Logarithm Function and Its Graph, we cover the following key points:

- What is Euler's Number? How do we find it?
- What is the history of Euler's Number?
- How do we write Euler's Number?
- What is a natural logarithm?
- What are the rules applied in natural logarithms?
- How do we solve equations involving natural logarithms?
- How do we deal with natural logarithm functions?
- How do we plot the graph of natural logarithm functions?
- How do we model the curves produced by natural logarithm functions?

Euler's Number e is an irrational number (as there is no recurrence in the order of the digits) obtained by finding the value of the number series

S∞ = __∞____∑___{n = 0}*1**/**n!* = *1**/**0!* + *1**/**1!* + *1**/**2!* + *1**/**3!* + *1**/**4!* + ⋯

The value of this series is e = 2.7182818284590

Euler's Number was first identified when trying to maximize the profit using the compound interest formula

A_{n} = P ∙ (1 + r)^{n}

At reducing time periods. The calculations resulted in the maximum value written above, which corresponds to the value of Euler's Series.

Another feature that makes Euler's Number special is the fact that it is the only number for which the derivative and integral have the same value - a value that corresponds to the original number itself.

When Euler's number represents the base of an exponential expression, we often write exp(x) instead of e^{x} ('exp' stands for 'exponential').

In simple words, Euler's number is the base of an exponential function whose rate of growth is always proportionate to its present (actual) value. The exponential function e^{x} always grows at a rate of e^{x} - a feature that is not true of other bases and one that vastly simplifies the algebra surrounding exponents and logarithms.

The natural logarithm of a number a is the logarithm with the base e of that number. We write 'ln' instead of 'log' to express the natural logarithm. (The symbol 'ln' is an acronym for 'natural logarithm' in Latin) When doing this, we do not write the base e, similar to the base 10 logarithms. Thus,

ln a = log_{e} a

The importance of the natural logarithm is evident in the fact that all calculators contain the 'ln' option.

The rules for natural logarithm are similar to those of standard logarithms given the fact that natural logarithm is a special type of logarithm. They are:

**Product rule**ln (x ∙ y) = ln x + ln y**Quotient rule**ln= ln x - ln y*x**/**y***Power rule**ln x^{n}= n ∙ ln x**Euler's Number raised to ln rule**e^{ln}x = x**Oher natural logarithm rules**- ln 1 = 0
- ln 0 is undefined (it is minus infinity)
- ln (-1) = i ∙ πwhere 'i' is the imaginary number where i = √(-1).

We can solve the equations involving natural logarithms in the same way as the rest of logarithmic functions. Thus, first we isolate the term(s) containing the variable, then we use any exponential transformation to isolate the variable only and eventually calculate its value. Sometimes, any of the roots in a natural logarithmic equation is not accepted, as the argument becomes negative for that value.

The natural logarithm function is a special type of logarithmic function. The simplest form of the natural logarithmic function is

y(x) = ln x

Other functions that contain the natural logarithm are

y(x) = k ∙ ln x = ln x^{k} ; y(x) = ln (-x) = -ln x; y(x) = ln (kx) = ln k + ln x; etc.

Obviously, we give some values to the independent variable x and by substituting them in the function's formula we then calculate the corresponding y-values, which represent the dependent variable.

The graph of 'ln' function is similar to that of the other logarithmic functions.

The natural logarithmic function y(x) = ln x is the inverse of the natural exponential function y(x) = e^{x}, in the sense that the graphs of these two functions mirror each other according the line y = x, which acts as a symmetry line. The symmetry is observed when one figure is folded according to the line of symmetry to give the other figure.

The procedure for modelling y = m · ln (kx) curves is similar to that used for modelling curves of the form

y = k ∙ a^{x}

According to this procedure, we first take the 'ln' off both sides to remove Euler's Number e; then, we calculate the values of the new variables. After this step is complete, we plot the linear graph obtained by replacing the old x- and y-values with the ln x and ln y ones in a ln y vs ln x graph. For this, the function is transformed in the following way:

y = a ∙ e^{b}x

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