Rules and Examples
log416=2⇆42=16
logbn=e⇆be=n
Given b=10:
n=10e/Rarrn is expressed as a function of e
log10n=e⇒e is expressed as a function of n
logx1=0⇒ always 0 regardless of the base because n0=1
log(10)=1⇒ lack of subscript implies a base of 10.
Notice how in 10x=n, the number of zeros in n is equal to x:
log(100)=2
log(1000)=3
log(1000000)=6
Likewise:
log(0.1)=−1
log(0.01)=−2
log(0.001)=−3
log(0.00001)=−5
As a rule in log(x), x>0 has to be true.
Further rules
- If a fraction is given to the log function, the result is always a negative number.
- If the number given is greater than the base, the result is always a whole number.
- If the base is greater than the given number, the result is always a fraction of a number < 1.
Change of Base Formula
loga(b)=logc(a)logc(b)
Since calculators often only support base 10 and ln for log functions, this formula can be used to solve log with different bases:
log4(16)= ? but since log(4)log(16)=2
Properties of Logarithms
log(A)+log(B)=log(AB)
log(A)−log(B)=log(BA)
log(A2)=2log(A)
Example 3 shows that logarithms can be used to algebraically find an unknown exponent in an equation:
x=y2⇒logx=2logx
log(x)+log(y)−log(z)=log(zxy)
Basically when adding and subtracting, positive logs move to the numerator and negative logs move to the denominator of a fraction.
Natural Log
Naturally arising logs in advanced mathematics and engineering are calculated to a base e given by the sequence
1+11+11.2+11.23+11.234+… to infinity
The sum of above sequence is e, which can be calculated to any degree of accuracy.
e.g. e=2.71828
The natural logarithm e is often abbreviated ln.
So loge10=ln10
Simplification
ln1=0
lne=1
lne5=5
When a logarithm is taken as an exponent, and the base is equal to the logarithm's base, they cancel and only the input to the logarithm remains:
eln7=7
Logarithmic Equations
Considering the exponential form often helps in solving equations involving logarithms.
logx27=3 solve for x
x3=27
x=327
x=3
When solving leads to multiple values for x, remember that the input x in log(x) can never be 0 or smaller.
Logarithm of a Root
Since logarithmic rules are derived from exponent laws it follows that
log4a=loga41
=41loga
Reciprocal
The logarithm of a number is equal to the negative of its reciprocal
log(91)=log1−log9
=0−log9
=−log9