### Change Of Base Formula In Logarithms

In the example below, let’s see what the value of the expression would be log_{2}(50) Ever since 50 does not represent a rational power of 2Without a calculator, it is difficult to estimate.

The majority of calculators, however, only implement logarithms directly in base-10 and base-*e*. For this reason, to find the value of log_{2}(50). The base of the logarithm must first be changed.

# Change in base Formula

Following is a rule that lets us change any logarithm’s base.

=

Applied to logs with a standard base, this rule states that evaluating one can be done by dividing by the standard base log of a non-standard base log. To keep it straight, I always look at the current position. Because the base is subscripted, the argument is in the original log above the base. So, when I split up the log, I left things that way.

Let us take an example here,

Log_{2}(4) =

=

Log_{2}(4) = 2.00000

It is best if you do the calculations entirely within your calculator since I showed the denominator and numerator values in the previous calculation. You won’t have to worry about writing it out.

If you want to minimize round-off errors, you should convert all the steps of division and evaluation into your calculator at once. To simplify calculating log(4) and log(2), rather than writing down the first eight decimal places of each value and dividing, just type in “log(4) / log(2)” into your calculator.

## Practice Exercises

#### On this topic, you may receive some simple (but rather useless) exercises. It’s easy; as long as you keep in your head the change-of-base formula, you’ll easily get it. Example:

#### · Convert the base of log_{2}(4) to the base of 3 and solve it again for your practice.

The only reason for a base-3 log must be for practice using change-of-base, so I don’t see any particular benefit to them.

Log_{2}(4) =

**Convert log(4) to the common log.**

If I already calculate the natural log in my calculator, why would I need to do this in the real world? No, this is just a practice exercise (and easy points).

Log(4) =

For exercises such as these, where the objective is converting using change-of-base and not getting actual decimal values, we should just leave the answer as a logarithmic fraction.

Despite the pointlessness of the above exercises, the change-of-base formula can be very useful in finding the plot points for non-standard logs whenever you have to use a graphing calculator to find them.

#### · Use the examples below for expressions to graph *y* = log_{3}(*x*).

However, I have used examples of both log_{2} and log_{3}, therefore, it will be helpful for you to understand better.

- since 2
^{-3}= , then log_{2}() = -3 - since 2
^{–2}= , then log_{2}(½) = –2 - since 3
^{0}= 1, then log_{3}(1) = 0 - since 3
^{1}= 3, then log_{3}(3) = 1 - since 3
^{2}= 9, then log_{3}(9) = 2 - since 3
^{3}= 27, then log_{3}(27) = 3 - since 3
^{4}= 81, then log_{3}(81) = 4

This is why I chose these particular x-values: anything smaller would have been impossible to graph by hand, and anything larger would have resulted in a ridiculously wide graph. I decided on the values that fit my needs.)

The graph should be drawn with my graphing calculator in this case. Where do I begin? If I wanted to plot some neat points using the “TABLE” feature on my graphing calculator, I don’t have a “log-base-two” button. However, I can convert the given function into something that my calculator understands thanks to the change-of-base formula.

But, I will leave this task to you guys. Do follow the exercise and try to draw graphs of the following values.