Logarithmic Function

LOGARITHMIC FUNCTION

DEFINITION AND FORM

A logarithmic function is the inverse function of an exponential function. It is written in the form:
f(x) = logₐ(x),
where 'a' is the base of the logarithm, a > 0, and a ≠ 1. The function is defined for x > 0, as the logarithm is not defined for zero or negative numbers in real numbers.

Mathematical interpretation:
If f(x) = logₐ(x), then this means:
a^(f(x)) = x,
which signifies that the logarithm tells us to what power the base 'a' must be raised to obtain the number 'x'.

DOMAIN AND RANGE

• DOMAIN (Df): (0, ∞) (positive real numbers)
• RANGE (Zf): ℝ (the values of the logarithm can be any real number)
• The function is continuous and strictly monotonic (either increasing or decreasing, depending on the base 'a').

FUNCTION BEHAVIOR BASED ON THE BASE

• If a > 1, the function is increasing.
• If 0 < a < 1, the function is decreasing.

In both cases, the following hold true:

• f(1) = logₐ(1) = 0
• lim (as x → 0⁺) logₐ(x) = –∞
• lim (as x → ∞) logₐ(x) = ∞ (if a > 1) or = –∞ (if 0 < a < 1)

GRAPHICAL PROPERTIES

• The function has a zero only at the point x = 1, where logₐ(1) = 0 always holds.
• It has a vertical asymptote at x = 0, as the logarithmic function approaches –∞ when x approaches 0 from the right.
• It is the inverse of the exponential function: if f(x) = logₐ(x), then f⁻¹(x) = aˣ.

COMMON EXAMPLES

• log(x) – common logarithm (base 10)
• ln(x) – natural logarithm (base e ≈ 2.718)

CALCULATION EXAMPLE

Let f(x) = log₂(x):

• f(1) = 0
• f(2) = 1
• f(8) = 3
• f(0.5) = –1

The function is increasing; all values of x > 0 have a corresponding logarithm.

CONCLUSION

The logarithmic function is an essential concept in mathematics, as it allows for solving equations with unknowns in the exponent and for dealing with inverse relationships between quantities. It accurately describes slow growth or decay and is the mirror image of the exponential function with respect to the line y = x. Due to its characteristics, it is indispensable in logarithmic equations, analysis, and modeling mathematical processes.