Logarithms Formula Sheet
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Definition :
The logarithm of a positive number is the power of the base that produces the number.
A logarithm whose base is 10 is called a common logarithm . It has the default form log N .
A logarithm whose base is e is called a natural logarithm . It has the default form ln N .
logb N  x if and only if b x  N
“logarithmic form”
“exponential form”
or
ln N  x if and only if e x  N
“logarithmic form”
“exponential form”
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Properties of Logarithms : (obtained by applying the definition)
1.
logb 1  0
or
ln1  0
“A logarithm of 1 equals 0.”
2.
logb b  1
or
ln e  1
“A logarithm of the base equals 1.”
3.
logb bn  n
or
ln en  n
“A logarithm of a power of the base
equals that power.”
4.
b logb N  N
or
e ln N  N
“A base, to a logarithm power with the
same base, equals the antilogarithm.”
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Laws of Logarithms : (used to expand or simplify/condense log expressions)
1.
logb MN  logb M  logb N
ln MN  ln M  ln N
Product/Sum Law
2.
M 
logb    logb M  logb N
N
M 
ln    ln M  ln N
N
Quotient/Difference Law
3.
logb xn  n logb x
4.
logb M  logb N if and only if
ln xn  n ln x
M  N
Power/Coefficient Law
Equal Logs Law
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Change-of-Base Formula :
logb N 
log a N
log a b
logb N 
log N
ln N

log b
ln b
where a is the new base
where changed to common log or natural log
Revised – BW0916