Logarithmic Functions

Logarithmic Functions

A logarithmic function is used to better model functions which increase at an exponential rate. It is the inverse of the exponential function and the general equation is:


The two main log functions that are used in methods is log to base of eulers number, the natural log and log to the base 10, where every time x increases by 1 the function increases by a factor of 10.

Sketching log functions

Here are a few guidelines to sketching logarithmic functions.

Number one


Number two


Number three is that we can’t take a log of a negative number or zero, since any number to the power of another number cannot equal zero or a negative number, you just get smaller fractions. Therefore the domain is only real positive numbers.

Number four

x\rightarrow 0, f(x) \rightarrow \infty^{-}

Number five is that the range includes all real numbers, which means the function can equal anything.

Number six is that it is only a one to function, which means that every point on the x axis corresponds to one and only one point on the y axis

If we follow these rules the basic function is not hard at all to sketch.




The rules of sketching the function this function are similar to the exponential function

In this equation h represents the shift along the x-axis and k the shift along the y-axis. So our known co-ordinates will face a small shift


(-1,\frac{1}{a}) \rightarrow (-1+h,\frac{1}{a}+k)

(0,1) \rightarrow (h,1+k)

(1,a) \rightarrow (1+h,a+k)


The graph can be reflected in the x-axis by placing a negative in front of the function to get:


The graph can be reflected in the y-axis by placing a negative in front of x in the function to get:


Since we can’t have the log of a negative number then all the values of x for this function must be negative.


Here we can see a number of natural logs graphed here. Increasing the value of the base will cause the function to increase at a greater rate for values of x greater than one and decrease at a greater rate for values less than one but greater than zero.

Here we can see the green function is a reflection of the blue function in the x axis and the red function is a reflection of the blue function in the x axis.

It is also possible for a log function to be reflected in both the x and y axis, if the negative is applied before the function and the x value.

If there are translations applied to the function as well, you must first reflect the function before you translate them.


The effect of dilation is similar to what happened with the exponential functions. If there is a value k in front of the function the dilation factor will be k from the x axis, the value will just be multiplied by k.

If there was a x coefficient to the value of k inside the log brackets then the dilation factor to the y axis will be one over k, the x-coefficient will be multiplied by one over k for a given value.


Lets now have a look at an example:


Which we can rewrite as


Straight away we can see that there is a reflection in the x-axis, dilation by a factor of three from the x axis, a half from the y-axis, but a shift of the graph five over two to the left and three down.

At this stage we need to have the shape of a negative log in our mind and from there we know it will have both a x and y intercept. A y-intercept due to the shift to the left, allowing for some negative values of x.

The asymptote will move:

x=0 \rightarrow x=-2.5

Then find the y intercept, let x=0



Given there is no dilation we can just draw the normal exponential function going through the y intercept of e plus 3, after it has been reflected in the x-axis.