Rules and Examples
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:
As a rule in log(x), x>0 has to be true.
- 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
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
Example 3 shows that logarithms can be used to algebraically find an unknown exponent in an equation:
Basically when adding and subtracting, positive logs move to the numerator and negative logs move to the denominator of a fraction.
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.
The natural logarithm e is often abbreviated ln.
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:
Considering the exponential form often helps in solving equations involving logarithms.
logx27=3 solve for x
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
The logarithm of a number is equal to the negative of its reciprocal