Exponential and Logarithmic Functions

Exponential Function

Exponential function is the function $e^x$ the number (approximately 2.718281828) such that the function $e^x$ is its own derivative . The function is often written as $\exp{(x)}$, especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics. Sometimes the term exponential function is used more generally for functions of the form $f(x)=cb^x$, where the base $b$ is any positive real number and $c$ is a constant.

Graph of $f(x)=e^x$

The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.

The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this that led Jacob Bernoulli in 1683 to the number $\displaystyle \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$, now known as e. Similarly, $\displaystyle \exp(x) = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}$ first given by Euler.

The exponential function $e^x$ can be characterized in a variety of equivalent ways. In particular it may be defined by the following power series: $\displaystyle e^x = \sum_{n = 0}^{\infty} {x^n \over n! } = 1 + x + {x^2 \over 2! } + {x^3 \over 3! } + \cdots$. From this definition, you can check that $e^x$ is its own derivative: $\displaystyle \frac{d}{dx} e^x = e^x$.

Logarithmic Functions

The logarithm of a number $x$ with respect to base $b$ is the exponent by which $b$ must be raised to yield $x$. In other words, the logarithm of $x$ to base $b$ is the solution $y$ to the equation $b^y = x$ . For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: $1000=10\cdot 10 \cdot 10 = 10^3$. More generally, if $x=b^y$, then $y$ is the logarithm of $x$ to base $b$, and is written $y=\log_b(x)$, so $\log_{10}(1000)=3$.

Plot of $\log_2(x)$

The graph of the logarithm to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates $(2,1)$, $(4,2)$, and $(8,3)$. For example, $\log_2(8)=3$, because $2^3=8$. The graph gets arbitrarily close to the y-axis, but does not meet or intersect it.

For $f(x)=e^x$, $g(x)=\log_e(x)$ is the inverse function of $f(x)$ and vice versa. The logarithm to base $b=10$ is called the common logarithm and has many applications in science and engineering. The natural logarithm has the constant e ($\approx 2.718$) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base $b=2$ and is prominent in computer science.