Identify Characteristics of Logarithmic Functions

Exploring the log graph

The graph of the log function given by $y=\log_b\left(x\right)$y=logb​(x).

When the base $b>1$b>1, the graph of the log function is a monotonically rising curve with an ever decreasing positive gradient (monotonically rising essentially means continuously rising all of the time, and an ever decreasing positive gradient means that the gradient, while never reaching zero, gradually and continuously decreases its positive steepness). 

Where $00<b<1 the graph is a monotonically falling curve with an ever increasing negative gradient. No other base is possible. 

The two instances are shown here for base $2$2 and base $\frac{1}{2}$12​.

The curve drawn in red is $y=\log_2x$y=log2​x. Note that the domain is given by $x>0$x>0, and the range includes all reals. As the $y$y-axis logarithms are essentially the specific indices of the powers of $2$2 required to form each independent $x$x, they grow slowly. For example, values of $x$x such as $x=1,2,4,8,16,32...$x=1,2,4,8,16,32...can be represented as powers of $2$2 by $2^1,2^2,2^3,2^4,2^5,...$21,22,23,24,25,... and thus the logs of these numbers are $y=1,2,3,4,5....$y=1,2,3,4,5....

Values of $x$x such as $x=\frac{1}{2},\frac{1}{4},\frac{1}{8}...$x=12​,14​,18​... will similarly have logs of $y=-1,-2,-3,...$y=−1,−2,−3,... etc so as $x$x gets closer and closer to $0$0, the logarithms of $x$x fall into a bottomless well. For example a value like $x=\frac{1}{8192}=2^{-13}$x=18192​=2−13 has a logarithm given by $y=-13$y=−13.  

The curve drawn in blue is $y=\log_{\frac{1}{2}}x$y=log12​​x. A nice way to understand the graph is to utilise the change of base rule. So:

$y$y	$=$=	$\log_{\frac{1}{2}}x$log12​​x
	$=$=	$\frac{\log_2x}{\log_2\left(\frac{1}{2}\right)}$log2​xlog2​(12​)​
	$=$=	$\frac{\log_2x}{-1}$log2​x−1​
	$=$=	$-\log_2x$−log2​x

This shows us that $y=\log_{\frac{1}{2}}x$y=log12​​x is the reflection of $y=\log_2x$y=log2​x, and the same idea would apply to any other base.

Note that both the red and the blue graphs pass through the point $\left(1,0\right)$(1,0). This is because any defined base raised to the power of $0$0 is $1$1, so that $\log_b1=0$logb​1=0.  

The Applet

The following log graph applet allows you to experiment with different bases. You should note that as the base increases beyond $1$1 the rate of increase in the size of the logarithm decreases.

As you move the base back again closer and closer to $1$1 from above, the rate increases so that the curve becomes more and more vertical. You should be able to see why the function cannot exist for bases equal to $1$1. 

For positive bases less than $1$1, try moving $b$b across the full range of values. What do you notice?

A special base 

There is one particular logarithm base that has become very important  in mathematics. Logarithms derived using this special base are known as natural logarithms. The base is the irrational number given by $e=2.7182818...$e=2.7182818... . The logarithms are called natural because $e$e is a number that arises in nature frequently and unavoidably. The base turns out to have far reaching ramifications in the study of such areas as the calculus, the mathematics of finance, sequences and series, and many other areas. In fact, the number $e$e is quite possibly the second most important constant in mathematics today. 

The function $y=\log_ex$y=loge​x is often written $y=\ln x$y=lnx, where the "ln" part is short for natural logarithm. The graph of this  particular function has the remarkable property that the gradient of any tangent drawn to it has a value equal to the inverse of the abscissae (the $x$x - value) of the tangent's point of contact, as depicted in this diagram. 

Try using the applet to create a logarithm function as close to $y=\ln x$y=lnx as you can. The applet increments in one-hundredths of units.  

Worked Examples

question 1

question 2

question 3