SOLUTION: In a scientific calculator the base for a lagarithm equation is 10 by default. How to change the base? Please explain elaborately.

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Question 474483: In a scientific calculator the base for a lagarithm equation is 10 by default. How to change the base? Please explain elaborately.
Found 2 solutions by Theo, richard1234:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the base change formula is very simple.
the formula is:
log%28b%2Cx%29+=+log%28c%2Cx%29+%2F+log%28c%2Cb%29
b is equal to the base you are converting from.
c is equal to the base you are converting to.
you are taking the log of x to the base of c and you are dividing that by the log of b to the base of c.
the bases you are converting to are most often the base of 10 or the base of e.
the base of 10 is the LOG function of your calculator.
the base of e is the LN function of your calculator.
If you have a logarithm that is other than to the base of 10 or the base of e, then you can solve it using your calculator by converting to the base of 10 or to the base of e.
note that log%2810%2Cx%29 is shown as log%28x%29.
note also that log%28e%2Cx%29 is shown as ln%28x%29.
to convert from any base to the base of 10, the formula is:
log%28b%2Cx%29+=+log%28x%29+%2F+log%2810%29
to convert from any base to the base of e, the formula is:
log%28b%2Cx%29+=+ln%28x%29+%2F+ln%2810%29
an example will clarify:
suppose you have a number such as log%282%2C64%29
you know that log%282%2C64%29 = 6 because you know that 2%5Ey = 64.
if you didn't know that, you would use the conversion formula to get it to the base of 10 so you can use the log function of your calculator, or you would usse the conversion formula to get it to the base of e so you can use the ln function of your calculator.
the conversion formulas are:
log%282%2C64%29+=+log%2864%29%2Flog%282%29+=+6
log%282%2C64%29+=+ln%2864%29%2Fln%282%29+=+6
note that:
log%282%2C64%29+=+log%2864%29%2Flog%282%29+=+6 is equivalent to log%282%2C64%29+=+log%2810%2C64%29%2Flog%2810%2C2%29+=+6
note also that:
log%282%2C64%29+=+ln%2864%29%2Fln%282%29+=+6 is equivalent to log%282%2C64%29+=+log%28e%2C64%29%2Flog%28e%2C2%29+=+6
a very short version of what I just said can be found here:
http://hyperphysics.phy-astr.gsu.edu/hbase/logm.html

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
I don't know if this requires an elaborate explanation (however it could get deep into proving the change-of-base formula then deriving the Taylor series for logarithms, then proving Taylor's theorem in the first place, etc.).

Simply put, if you want to convert from base b to base 10 (so you can input it on your calculator), use the change-of-base formula:



Then it is easy to input it on a calculator.



Also, most graphing calculators can allow logarithms of bases other than 10 or e. They are quite expensive, though, but graphing calculators can do all sorts of functions such as sums/products, derivatives/integrals, limits, determinants of matrices, factorials, defining your own functions, etc.