There is one other log "rule", but it's more of a formula than a rule.
You may have noticed that your calculator only has keys for figuring the values for the common (that is, the base-10) log and the natural (that is, the base-e) log. There are no keys for any other bases. Some students try to get around this by "evaluating" something like "log_{3}(6)" with the following keystrokes:
[LOG] [ 3 ] [ ( ] [ 6 ] [ ) ]
Of course, they then get the wrong answer, because the above actually (usually) calculates the value of "log_{10}(3) × 6". This is not what had been intended.
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In order to evaluate a non-standard-base log, you have to use the Change-of-Base formula:
Change-of-Base Formula:
What this rule says, in practical terms, is that you can evaluate a non-standard-base log by converting it to the fraction of the form "(standard-base log of the argument) divided by (same-standard-base log of the non-standard-base)". I keep this straight by looking at the position of things. In the original log, the argument is "above" the base (since the base is subscripted), so I leave things that way when I split them up:
Here's a simple example of this formula's application:
The argument is 6 and the base is 3. I'll plug them into the change-of-base formula, using the natural log as my new-base log:
Then the answer, rounded to three decimal places, is:
log_{3}(6) = 1.631
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I would have gotten the same final answer if I had used the common log instead of the natural log, though the numerator and denominator of the intermediate fraction would have been different from what I displayed above:
As you can see, it doesn't matter which standard-base log you use, as long as you use the same base for both the numerator and the denominator.
While I showed the numerator and denominator values in the above calculations, it is actually best to do the calculations entirely within your calculator. You don't need to bother with writing out that intermediate step.
In fact, to minimize on round-off errors, it is best to try to do all the steps for the division and evaluation in your calculator, all in one go. In the above computation, rather than writing down the first eight or so decimal places in the values of ln(6) and ln(3) and then dividing, you would just do "ln(6) ÷ ln(3)" in your calculator.
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You may get some simple (but fairly useless) exercises on this topic. Don't begrudge them; they're easy points, as long as you keep the change-of-base formula straight in your head. For instance:
I can't think of any particular reason why a base-5 log might be useful, so I think the only point of these problems is to give you practice using change-of-base. Fine; I'll plug-n-chug:
Why on earth would I want to do this (in "real life"), since I can already evaluate the natural log in my calculator? I wouldn't; this exercise is just for practice (and easy points).
I'll plug-n-chug into the change-of-base formula:
Since getting an actual decimal value is not the point in exercises of this sort (the converting using change-of-base is the point), just leave the answer as a logarithmic fraction.
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While the above exercises were fairly pointless, using the change-of-base formula can be very handy for finding plot-points when graphing non-standard logs, especially when you are supposed to be using a graphing calculator.
If I were working by hand, I would use the definition of logs to note that:
And then I would draw my graph by hand.
(Why did I pick these particular x-values? Because anything smaller would have been too tiny to graph by hand, and anything larger would have led to a ridiculously wide graph. I picked the values that fit my needs.)
But, in this case, I'm supposed to be doing the graph with my graphing calculator. How can I do this? (Or what if I'd just like to use my graphing calculator's "TABLE" feature to find some nice neat plot points?) I don't have a "log-base-two" button. However, I can enter the given function into my calculator by using the change-of-base formula to convert the original function to something that's stated in terms of a base that my calculator can understand. Flipping a coin, I choose the natural log:
(I could have used the common log, too. In that case, the function would have been "y1 = log(x)/log(2)".)
In my graphing calculator, after adjusting the viewing window to show useful parts of the plane, the graph will look something like this:
By the way, you can check that the graph contains the expected "neat" points (that is, the points I would have calculated by hand, as shown above) to verify that the picture displays the correct graph:
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