Unizor - Algebra - Logarithmic Functions - Graph скачать в хорошем качестве

Unizor - Algebra - Logarithmic Functions - Graph

The main basis for graphing a logarithmic function is the fact that it is an inverse of an exponential function with the same base. As you know, an exponential function Y = a^X with a base a greater than 1 is monotonic and increases from non-inclusive 0 to positive infinity as its argument X increases from negative infinity to positive infinity. Its graph reflects this behavior and passes on its way through a key point . The corresponding logarithmic function Y = LOGaX with the same base a is greater 1 is inverse to this exponential function. Therefore, the graph of this logarithmic function is symmetrical to the graph of a corresponding exponential function relative to a bisector of the angle between X-axis and Y-axis (see discussion about graphs of inverse functions in a section Math Concepts - Functions - Monotonic). This graph would have its argument X changing from non-inclusive 0 to positive infinity (compare to a range of an exponential function) with the function values Y monotonously increasing from negative infinity, crossing zero value when the argument X = 1 and further increasing to positive infinity (compare to a domain of an exponential function). Let's analyze the dependency of a graph of a logarithmic function from the value of its base a is greater than 1. As we know from the properties of the exponential function Y = a^X with these bases, greater value of a base results in a steeper graph as the argument X moves to positive infinity and also results in a graph positioned closer to the X-axis (that is, smaller function value) as the argument moves to negative infinity. Therefore, the graph of the corresponding inverse logarithmic function Y = LOGaX with the greater value of a base a is greater 1 would be closer to the Y-axis decreasing to negative infinity as its argument approaches 0 and less steeply increasing to positive infinity as its argument increases to positive infinity. Algebraically, if we compare two logarithmic functions Y = LOGaX and Y = LOGbX, where 1 is less than a and a is less than b, for different values of an argument X, we can see that for X greater than 1 (where both logarithms are positive) greater base causes smaller value of the logarithm (that is LOGaX is greater than LOGbX) and for interval (0,1) (where both logarithms are negative) greater base causes greater (smaller by absolute value) value of the logarithm (that is LOGaX is less than LOGbX). Proofs of these statements are in the video lecture.