Logarithmic function

Logarithm with base $a$. The logarithm of $x$ with base $a$, denoted by $ \log_a x$, is defined as the unique value $ u$ satisfying $ x = a^u$. In other word we have

$\displaystyle y = \log_a x$    if and only if $\displaystyle \quad x = a^y .$ (1)

Immediate properties of logarithms are as follows:

  $\displaystyle \log_a 1 = 0$   $\displaystyle \mbox{(since $a^u = a^0 = 1$); }$    
  $\displaystyle \log_a a = 1$   $\displaystyle \mbox{(since $a^u = a^1 = a$); }$    
  $\displaystyle \log_a a^b = b$   $\displaystyle \mbox{(since $a^u = a^b = a^b$); }$    
  $\displaystyle a^{\log_a b} = b$   $\displaystyle \mbox{(since $a^u = b$ where $u = \log_a b$). }$    

Common and natural logarithms. The common logarithm of $x$ (denoted by $ \log x$) and the natural logarithm of $x$ (denoted by $ \ln x$) are defined respectively by

$\displaystyle \log x = \log_{10} x
$

and

$\displaystyle \ln x = \log_e x$   (where $ e \approx 2.71828\ldots$)

Logarithmic function. Let $ a \neq 1$ be a positive real value. A function $f$ is called an logarithmic function with base $a$ if

$\displaystyle f(x) = \log_a x .
$

When $ a > 1$, $f$ is an increasing function and $ f(x) \to -\infty$ as $ x \to 0+$. When $ 0 < a < 1$, $f$ is a decreasing function and $ f(x) \to \infty$ as $ x \to 0+$. In either case, the logarithmic function $f$ has the vertical asymptote $ x = 0$. The domain $ D$ and the range $ R$ of $f$ are given by $ D = (0, \infty)$ and $ R = (-\infty, \infty)$, respectively.

1. $ a > 1$   2. $ 0 < a < 1$
\includegraphics{lec08a.ps}   \includegraphics{lec08b.ps}

Properties of logarithmic functions. The logarithmic function  $ f(x) = \log_a x$ is a one-to-one function; thus, $ \log_a x_1 = \log_a x_2$ implies $ x_1 = x_2$. Furthermore, (2) indicates that $ f(x) = \log_a x$ is the inverse function $ g^{-1}(x)$ of the exponential function  $ g(x) = a^x$.

Natural logarithmic function. A function $f$ is called the natural logarithmic function if

$\displaystyle f(x) = \ln x .
$



Department of Mathematics
Last modified: 2005-09-29