Master logarithms with this comprehensive guide! Learn the definition, properties (product, quotient, power rules), change of base, and solving logarithmic equations. Includes examples, cologarithm, antilogarithm, and inequalities. Perfect for students and math enthusiasts!
Definition
Tip
The logarithm of a number a>0 in base b (b>0, b=1) is the exponent c to which b must be raised to obtain a:
logba=c⇔bc=a
Logarithms in the Real Numbers
Domain: a∈(0,+∞)
Range: c∈R
Restrictions:
b>0,b=1
General Properties of Logarithms
Important
The logarithm of the base equals one:
logbb=1
Examples
log22=1
log1010=1
log55=1
logee=1
log100100=1
log33=1
The logarithm of 1 in any base is zero:
logb1=0
Examples
log21=0
log101=0
log51=0
loge1=0 (i.e., ln1=0)
log1001=0
log31=0
Logarithm of a product in the same base:
logb(xy)=logbx+logby
Examples
log2(4⋅8)=log24+log28
log3(9⋅27)=log39+log327
log10(5⋅2)=log105+log102
ln(e⋅e2)=lne+lne2
log5(25⋅125)=log525+log5125
log6(6⋅36)=log66+log636
Logarithm of a quotient in the same base:
logb(yx)=logbx−logby
Examples
log2(28)=log28−log22
log3(927)=log327−log39
log10(101000)=log101000−log1010
ln(e2e5)=lne5−lne2
log5(25125)=log5125−log525
log6(636)=log636−log66
Logarithm of a power:
logb(xn)=nlogbx
Examples
log2(43)=3log24
log3(92)=2log39
log10(1004)=4log10100
ln(e7)=7lne
log5(253)=3log525
log6(65)=5log66
Logarithm of a root:
logbnx=n1logbx
Examples
log238=31log28
log39=21log39
log10410000=41log1010000
ln5e10=51lne10
log53125=31log5125
log636=21log636
Logarithm with exponential base and argument:
logbmxn=mnlogbx
Examples
log2382=32log28
log1021004=24log10100
log3492=42log39
loge5e7=57lne
log52253=23log525
log6663=63log66
Equivalence of logarithmic expressions:
logbx=logbnxn=logmbmx
Chain rule:
logby⋅logya⋅logax=logbx
Examples
log24⋅log48⋅log816=log216
log39⋅log927⋅log273=log33
log10100⋅log1001000⋅log100010=log1010
log525⋅log25125⋅log125625=log5625
ln2⋅log2e⋅loge4=ln4
log636⋅log366⋅log6216=log6216
Unit product:
logbx⋅logxb=1
logbx=logxb1
Change of base:
logba=logkblogka(k>0,k=1)
Exchange rule:
xlogby=ylogbx
Special properties:
blogbx=x
Cologarithm
Tip
Defined as the logarithm of the reciprocal of a number: