Precalculus Review

Logarithmic Functions

Logarithm with base a. The logarithm of x with base a, denoted by $\log_a x$ , is defined as the unique value y satisfying

. In other word we have

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

Immediate properties of logarithms are as follows:

$\log_a 1 = 0$ (since );
$\log_a a = 1$ (since );
$\log_a a^b = b$ (since );
$a^{\log_a b} = b$ (by definition we have where $y = \log_a b$ ).

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 $e \approx 2.71828\ldots$ ) $\displaystyle$

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

, f is a decreasing function and $f(x) \to \infty$ as $x \to 0+$ . In either case, the logarithmic function

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.	2.
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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 . Furthermore, (2) indicates that $f(x) = \log_a x$ is the inverse function $g^{-1}(x)$ of the exponential function .

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

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