Recall that a function of the form y = f (x) = a^x; a > 0, a ≠ 1 is called an exponential function. Its domain is the set of all real numbers. In this sub-unit, we introduce logarithmic functions.

More precisely logarithmic function can be defined as follows:

Domain of logarithmic function is the set of all positive real numbers. That is, domain of logarithmic function is {x∊ℝ:x>0} and its range is the set of all real numbers, ℝ.

The expression y = log_ax is equivalent to a^y = x and indicates that the logarithm y is the exponent to which b must be raised to obtain x. The expression y = log_ax is called the logarithmic form of the equation, and the expression a^y = x is called the exponential form of the equation.

Basic Properties of the Logarithmic Functions f (x) = log_b x; b > 0 and b ≠ 1

1. The domain is the set of all positive real numbers, {x∊ℝ : x > 0}.
2. The range is the set of all real numbers, ℝ.
3. The graph intersects the x-axis at the point (1, 0). Thus, the x-intercept is (1, 0).
4. The graph never cross or touch the y-axis. Thus, there are no y-intercepts.
5. The graph is smooth curves and continuous with no corners and gaps.
6. If b > 1, then the logarithmic function is an increasing function.

Logarithmic equations

An equation is called logarithmic equation, if the variable is part of a number whose logarithm is to be taken.

In this section, we will see how to give solutions of equations which involve logarithms by applying the properties studied.

In solving logarithmic equation, note that if log_a x = 0, then x = 1. In solving any log equation you note that x > 0.