# Natural Logarithm

## Natural Logarithmic Function

The natural logarithm function $y = ln(x)$ is the logarithm with the base equal to the Euler Constant e and is therefore the inverse function of the natural exponential function $y = e^x$. Hence
$y = ln(x) \;\;\; \text{if and only if} \;\;\; x = e^y$
The graphs of $y = ln(x)$ and its inverse $y = e^x$ are shown below. Each of the two graphs is a reflection of the second on the line y = x because they are inverse of each other.

In general the composition of a function $f$ and its inverse $f^{-1}$ are related by:
1) $f(f^{-1}(x)) = x$, x in the domain of $f^{-1}$
and
2)$f^{-1}(f(x)) = x$, x in the domain of f.
Hence
Since $ln(x)$ and $e^x$ are inverse of each other, we have $ln(e^x) = x$ and $e^{ln(x)} = x , x \gt 0$ .
Numerical examples
$ln(e^3) = 3$
$ln(1) = ln(e^0) = 0$
$ln(e) = ln(e^1) = 1$
$ln(1/e) = ln(e^{-1}) = - 1$
$ln(\sqrt e) = ln(e^{1/2}) = 1/2$

## Properties and Rules of the Natural Logarithmic Functions

Domain: $(0 , +\infty)$
Range: $(-\infty , +\infty)$
x inttercept $(1,0)$
Vertical Asymptote: $x = 0$ because $\lim_{x \to 0^{+}} \ln(x) = - \infty$
Monotonicity: increasing on the interval $(0 , +\infty)$
Continuity: continuous on the interval $(0 , +\infty)$
Differentiability: differentiable on the interval $(0 , +\infty)$
One to One Function: It is a one to one function
Inverse Function: The inverse of the natural logarithm function $f(x) = ln(x)$ is $g(x) = e^x)$
Composition with Inverse: $\ln(e^x) = x$ and $e^{\ln(x)} = x$ for $x \gt 0$
Derivative: $\dfrac{d \; ln(x)}{dx} = \dfrac{1}{x}$

Rules of the natural logarithm ln(x)
$ln(a \times b) = ln(a) + ln(b)$ , for a anb b positive
$ln(\dfrac{a}{b}) = ln(a) - ln(b)$ , for a anb b positive
$n \; ln(x) = ln(x^n)$ , for x positive

A logarithm to any base may be written as a natural logarithm using the change of base formula
$log_a(x) = \dfrac{ln(x)}{ln(a)}$

## Mathematical Models Using Exponential Function

Natural logarithmic functions are used to solve equations related to mathematical models including natural exponential functions.

Example 1: Population Growhth Modeling
The population P of a small city varies continously according to the formula
$P = 10000 e^{0.025 t}$
where t is the number of years after the year 2019.
When will be the popultaion reach 12000?

Solution to Example 1
We know the population and we want to find t.
Hence the equation
$12000 = 10000 e^{0.025 t}$
Divide both sides of the equation by 10000 and simplify
$1.2 = e^{0.025 t}$
take the natural logarithm of both sides
$ln(1.2) = ln(e^{0.025 t})$
Use the property $ln(e^x) = x$ to simplify the right side of the equation
$ln(1.2) = 0.025 t$
Divide both sides of the equation by 0.025 and evaluate t
$t = ln(1.2) / 0.025 = 7.29286 years$
We need to round t = 7.29286 years to the nearest integer, hence
$2019 + 7 = 2026$
The population of the city will reach 12000 in the year 2026.

Example 2: Continuous Compounding Interest
The balance B in a saving account continously compounded is given by
$B = P e^{r t}$
where r is the annual rate of interest and $P$ is the principal.
An amout of $\ 50000$ is invested at the rate $r = 5.5\%$ and compounded continuously. How many years, after the initial deposit of $\50000$, will it take for the investment to reach $\75000$?

Solution to Example 2
Let t = 0 when the initial $\50000$ is deposited. Hence the balance B is given by
$B = 50000 e^{0.055 t}$
we need to find t when B is equal to \$75000.
$75000 = 50000 e^{0.055 t}$
divide both sides of the equation by 50000 and simplify to obtain
$1.5 = e^{0.055 t}$
take natural logarithm of both sides
$ln( 1.5 ) = ln (e^{0.055 t})$
Use the property $\ln(e^x) = x$ to simplify right side of the equation and
$0.055 t = ln(1.5)$
$t = ln(1.5) / 0.055 \approx 7.4$ years