Logarithms
Logarithms are inverse operations of exponentiation.
For example, since 24 = 16,
log216 = 4
Similarly, log2 of 8 is 3.
Raising a number to a positive power increases the number, whereas a logarithm of a number decreases the number.
Properties
If we take the log of an exponential, and the bases are the same, then the output is the exponent. More precisely:
logb(bp) = p
For example:
log2(23) = 3
We can confirm this by evaluating the operation on the left. We know that:
23 = 8
We also know that:
log28 = 3
Thus,
log2(23) = 3
Here are some more examples:
b(logbx) = x
2(log28) = 8
23 = 8
property 2:
logb(x ⋅ y) = (logbx) + (logby)
log2(8 ⋅ 16) = (log28) + (log216)
log2(8 ⋅ 16) = 3 + 4
log2(8 ⋅ 16) = 7
This was useful when it was hard to compute x*y. We can take the log of both sides with base 10. Then we can compute the right side by consulting a log table, then adding the logs. Then we can use a "anti-log" table, which tells us how to compute 10x.
For example, suppose we want to calculate 14*18.
Then first take the log of it:
log10(14 ⋅ 18)
This is equal to
(log1014) + (log1018)
Now we can look up the log table. It tells us that
log1014 = 1.146
log1018 = 1.255
Then we add them:
2.401
Finally, we look up the anti-log table which tells us:
102.401 = 251.77
14*18 is actually 252, so this was quite close.
Property 3
logb(xp) = p ⋅ (logbx)
log2(83) = 3 ⋅ (log28) == 3*3 == 9
log2512 = 9
Log of Power property:
logb(bp) = p
log2(23) = 3
Changing the Base:
logbx = (logkx) / (logkb)
log28 = (log108) / (log102)
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