Lesson CHANGING THE BASE OF A LOGARITHM

This lesson covers an overview of CHANGING THE BASE OF A LOGARITHM

REFERENCES

http://www.sosmath.com/algebra/logs/log1/log1.html
http://www.themathpage.com/aPreCalc/logarithms.htm#laws
http://oakroadsystems.com/math/loglaws.htm
http://hyperphysics.phy-astr.gsu.edu/Hbase/logm.html
http://www.uncwil.edu/courses/mat111hb/EandL/logprop/logprop.html
http://mathforum.org/library/drmath/view/70648.html

VALUE OF CHANGING THE BASE OF LOGARITHMS

Being able to change the base of a logarithm is essential to solving problems involving bases other than the bases used in your calculator.

Most scientific calculators can solve logarithms to the base of 10 and to the base e.
Logs to the base 10 use the LOG function of the calculator.
Logs to the base e use the LN function of the calculator.

When someone says to take the log of something and doesn't specify the base, the base 10 is assumed.
When someone says to take the natural log of something, the base e is assumed.
All other logs must have the base specified.

Here's some examples:

Find the log of 3 (base 10 is implied).
Find the natural log of 3 (base e is implied).
Find the log of 3 to the base 2 (base 2 is explicitly declared).

Log of x can be written as log(x).
Log of x can also be written as log(10,x)
In the first form, base 10 is implied.
In the second form, base 10 is explicitly declared.

Natural log of x can be written as ln(x).
Natural log of x can also be written as log(e,x)
In the first form, base e is implied.
In the second form, base e is explicitly declared.

Log of to any base other than 10 or e must be explicitly declared in the form of log(b,x).

In the explicitly declared form of log(b,x), b is the base and x is the number that you want to get the logarithm of.

The basic definition of logarithms states:

y = log(b,x) if and only if b^y = x

Example:

5 = log(3,243) if and only if 3^5 = 243

You can use your calculator to confirm that 3^5 = 243.

You can't use your calculator to confirm that log(3,243) = 5.
Not directly, anyway.

You can, however, use your calculator to confirm that log(3,243) = 5 by using the formula that allows you to change the base of the logarithm.

That formula is:

Log(b,x) = log(c,x) / log(c,b)

b is the base you want to convert from.
c is the base you want to convert to.

The bases you will want to convert to most of the time are the bases that your calculator can handle.

Those are 10 and e.

10 is the LOG function of your calculator.
e is the LN function of your calculator.

Here's how you do it using your calculator.

Log(b,x) = LOG(x)/LOG(b)
Log(b,x) = LN(x)/LN(b)

Example:

We stated that log(3,243) = 5 if and only if 3^5 = 243

We were able to use our calculator to determine that 3^5 really is 243 because the exponential function of the calculator allows us to do that.

Now, using our conversion formula, we can determine that log(3,243) = 5.

We simply use the built in log functions as we normally would and divide the result by the log of the base we want to convert from.

Log(3,243) = LOG(243) / LOG(3) = 5
Log(3,243) = LN(243) / LN(3) = 5

Use your calculator to see that the result will be 5 in both cases.

LOG(243)/LOG(3) is equivalent to log(10,243) / log(10,3)
LN(243)/LN(3) is equivalent to log(e,243) / log(e,3)

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