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You are here: Home / Category Index / Algebra / Indices And Logarithms
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Indices And Logarithms - By Gordon Syme
How to use this applet:Use the pull-down menu to select a topic. For full instructions on how to use the 'Give it a go' page see Other Information.Notes on the maths used in the applet:Laws of Indices:
| 1) aman = am+n |
5) (a/b)m = am/bm |
| 2) am/an = am-n |
6) a0 = 1 |
| 3) (am)n = amn |
7) a-m = 1/(am) |
| 4) (ab)m = ambm |
8) am/n = nÖam |
Numbers written in Index form cannot be added or subtracted explicitly, but they are much easier to multiply and divide when they have the same base. As the laws show, multiplication of indices with the same base is achieved simply by adding the indices, while division of indices with the same base is achieved by subtracting the indices. An index may also be raised to a power, in which case you multiply the indices. Any number to the power of 0 is equal to 1.
Laws of Logarithms:
| 1) logam + logan = logamn |
4) logam = logcm/logca |
| 2) logam - logan = logam/n |
5) logaa = 1 |
| 3) logam = n.logam |
6) loga1 = 0 |
Since logarithms are an alternative method of writing indices, it can be shown that the laws of logarithms are equivalent to the corresponding laws of indices. Important things to note are that the logaa is equal to 1, and loga1 is equal to 0.
Other useful information: Instructions for using the applet:
Valid expressions are those that are shown to the left of the ‘=’ in the laws of Indices and Logarithms. The applet will show the semi-simplified version of the expression and then a completely simplified version.
To enter an Index use the notation x^y, where x is the base and y is the index.
Allowed index expressions are (shown in bold):
x^y = xy
x^y*x^z = xy*xz
x^y/x^z = xy/xz
(a*b)^z = (a*b)z
(a/b)^z = (a/b)z
(a^x)^z = (ax)z
x^(y/z) = x(y/z)
To enter a Logarithm use the notation logX(Y) where X is the base and Y is the number.
Allowed logarithm expressions are (shown in bold):
logX(Y)+logX(Z) = logx(y) + logx(z)
logX(Y)-logX(Z) = logx(y) - logx(z)
logX(YZ) = logx(yz)
logX(Y) C (Change of base expression - see below)
To specify that you want to change the base of a Logarithm, enter logX(Y) C, where X is the base, Y is the number and C is the new base that you want the Logarithm changed to.
Note that the only expression where a space is used is in the change of base Logarithm expression between the closing bracket and the new base.
Valid operators are + - * /. Also some operators will not be valid in certain situations, e.g. if you enter an expression that isn’t covered by any of the laws of Logarithms or Indices. For example: 2^6+2^3. This is not a valid expression because the laws of Indices only cover multiplying and dividing two indices that have the same base.
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