When does log(x-3) = log x – log 3?
[26 Oct 2013]
I recently had a Twitter exchange with Pat Ballew who tweets using the handle @OnThisDayinMath.
I’ve never liked aspects of logarithm notation, and my tweet that day pointed out that despite having this convention in math:
7(x − 3) = 7x − 21,
it happens that
log(x − 3)
is mostly NOT equal to
log(x) − log(3).
Such inconsistencies in math notation cause unnecessary confusion for new learners. (See more on this issue at Towards more meaningful math notation.)
Here’s the Twitter exchange:
My reply contained a link to a Desmos graph showing where the graphs of y = log(x − 3) and y = log(x) − log(3) intersect.
You only really understand many math problems if you can do them graphically, algebraically, and numerically. But I get most understanding from a graph.
First let’s see a graphical solution for this problem.
One way of solving equations is to graph the left and right sides, and look for where the graphs intersect, since that gives us the solution.
The graph I pointed to in the link was something like the following, where the green curve is y = log(x − 3) and the blue curve is y = log(x) − log(3):
Zooming in on the intersection point, we can see the 2 curves cross when x = 4.5, and the y-value is about 0.4 (actually, it’s loge(1.5) = 0.405465…).
I answered Pat’s question with a link to a graph as I felt that trying to explain the solution algebraically within the limitations of a 140-character tweet would be somewhat problematic.
Graphs a mystery?
I’ve been thinking about Pat’s comment that graphs "still leave a mystery here".
I spent most of my student days and early teaching life solving equations algebraically. That’s what most textbooks did, and that’s what the curriculum required, so we just went ahead and did it that way. We didn’t have access to computer graphing tools (or calculators early on), so there wasn’t much option – algebraic solutions were quicker than drawing graphs by hand or finding numerical solutions.
Most of the time I didn’t have much of a "feel" for what was going on when the equations became more complicated.
I certainly had a feel for the solution when solving linear, quadratic and other polynomial equations. If the number of terms in the polynomial was small, and it was a "nice" polynomial equation where the number of roots was equal to the degree of the polynomial, all was fine.
But then as things became more complicated, you needed a better understanding of what was happening so you wouldn’t miss some of the possible x-values. For me, that understanding came from graphs.
For example, once we started doing trigonometry I didn’t understand why there would be an infinite number of solutions for this equation:
sin 3x = 0.47
However, once I saw the graph of y = sin 3x, and noted where it intersected with the graph of y = 0.47, it was immediately obvious there were many solutions.
Back to the log question. Let’s now see how to solve our equation algebraically.
log(x − 3) = log x − log 3
Using the 2nd log law on this page, we know we can write the difference between 2 logs as:
log a − log b = log a/b
So the right hand side of our equation can be written:
log x − log 3 = log (x/3)
Our equation has become:
log(x − 3) = log (x/3)
Since there are 2 log expressions with the same base that are equal to each other, we know the expressions in brackets must be equal, so we have:
x − 3 = x/3
3x − 9 = x
2x = 9
x = 4.5
So assuming base e, the following is true:
log(4.5 − 3) = log 4.5 − log 3 = log 1.5 = 0.405465…
We set up a table of values as follows.
If we try any negative value of x, we get an error. Similarly, for log(x – 3), positive values of x give us trouble unless x is bigger than 3.
|x||log (x − 3)||log x − log 3|
It looks like we have found the only solution, x = 4.5.
The problem with numerical methods, like a lot of algebraic methods, is we are left wondering if there are any more solutions. You often get caught by not considering other values.
We’ve now solved the equation graphically, algebraically and numerically.
I find I get the best insights into the problem when I can see it, and so most of the time these days I choose to solve equations by drawing a graph (using computer graphing mostly.)
It’s quick, and it gives me the best feel for the meaning of what I’ve found.
(1) Different log bases
Does it matter which logarithm base we are using? Do we get the same x-value?
Above I was using base e (the natural logarithms). Here is the graph if we are using base 10.
We can see the intersection point is when x = 4.5, but the resulting y-value is lower.
Base e and Base 10 solution comparison
In the following graph, we can also see where the base e graphs intersect (in lighter colors) and the log (base e) value is shown with a red dot.
(2) Alternative graphical approach
We could have also approached this problem by getting everything on the left side of the equation, as follows:
log(x − 3) = log x − log 3, when
log(x − 3) − (log x − log 3) = 0
log(x − 3) − log x + log 3 = 0
Here’s the graph of y = log(x − 3) − log x + log 3
We can see it cuts the x-axis at x = 4.5, and this is our solution.
But this doesn’t give us as much "feel" about what’s going on as the graphs given at the beginning of this article, where we graphed both sides of the equaiton separately and then looked for the intersection.
(3) ln x or loge x?
I think “ln x” is terrible notation for natural logs. I have used loge x throughout this article as it is much easier to identify that it is a logarithm expression, and so easier to understand. I ranted about this issue in Logarithms – a visual introduction.