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The Change-of-Base Formula (page 5 of 5)

Sections: Basic log rules, Expanding, Simplifying, Trick questions, Change-of-Base formula

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 (base-10) log and the natural (base-e) log, but no other bases. Some students try to get around this by "evaluating" something like "log3(6)" with the following keystrokes:

    [LOG]   [ 3 ]   [ ( ]    [ 6 ]   [ ) ]

Of course, they get the wrong answer, because the above actually calculates the value of
log10(3) 6". In order to evaluate a non-standard-base log, you have to use the Change-of-Base formula:

    log_b(x) = log_d(x) / log_d(b)

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:

    argument 'x' in top log, base 'b' in bottom log

This is how you would evaluate the last example on the previous page:

  • Evaluate log3(6).

    The argument is 6 and the base is 3. I'll plug them into change-of-base, using the natural log as my new log:

      log_3(6) = ln(6) / ln(3) = 1.6309 (approx.)

    Then the answer, rounded to three decimal places, is:

      log3(6) = 1.631

You would get the same answer if you used the common log, though the numerator and denominator of the intermediate fraction would be different from what I did above:

    log_3(6) = log(6) / log(3) = 1.6309 (approx.)

As you can see, it doesn't matter which standard-base log you use, as long as you use the same base for the numerator and denominator.   Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

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 writing out the intermediate step. To minimize on round-off errors, 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.

You may also get some simple (but useless) exercises on this topic, such as:




  • Convert log3(6) to base 5.

    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:

      log_3(6) = log_5(6) / log_5(3)

  • Convert ln(4) to an expression written in terms of the common log.

    Again, why would you do this (in "real life"), since you can already evaluate the natural log in your calculator? You wouldn't; this exercise is just for practice:

      ln(4) = log(4) / log(e)

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.

On the other hand, using change-of-base is handy for finding plot-points when graphing non-standard logs, especially when you are supposed to be using a graphing calculator.

  • Use your graphing utility to graph y = log2(x).

    If you were working by hand, you would use the definition of logs to note that:

    • since 21 = 1/2, then log2(1/2) = 1
    • since 20 = 1, then log2(1) = 0
    • since 21 = 2, then log2(2) = 1
    • since 22 = 4, then log2(4) = 2
    • since 23 = 8, then log2(8) = 3
    • since 24 = 16, then log2(16) = 4

    And then you would draw the graph by hand. But what if you're supposed to do the graph in your calculator? (Or what if you'd like to use your graphing calculator's "TABLE" feature to find nice neat plot points?) You don't have a "log-base-two" button; instead, you can enter the function by using the change-of-base formula to convert to a base your calculator can understand:

      y1 = ln(x) / ln(2)

    The graph would like something like this:

      graph of y = log_2(x)

By the way, you can check that the graph contains the expected "neat" points (that is, the points you would have calculated by hand, as shown above) to verify that the picture displays the correct graph:

      graph with &quo

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Cite this article as:

Stapel, Elizabeth. "The Change-of-Base Formula." Purplemath. Available from Accessed


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