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The logarithm of a product can be written as the sum of the individual logarithms of each factor.
log_b mn=log_b m+log_b n
This property is only valid for positive values of b, m, and n, and for b≠ 1. As an example, the expression log_3 (7*4) can be rewritten using this property. log_3 (7*4)=log_3 7+log_3 4
Rewrite mn as m* n
m=b^(log_b(m))
a^m*a^n=a^(m+n)
log_b(b^m)=m
The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.
log_b m/n=log_b m -log_b n
This property is valid for positive values of b, m, and n, and for b≠ 1. For example, the expression log_3 74 can be rewritten using this property. log_3 7/4=log_3 7-log_3 4
m=b^(log_b(m))
a^m/a^n= a^(m-n)
log_b(b^m)=m
The logarithm of a power can be written as the product of the exponent and the logarithm of the base.
log_b m^n =nlog_b m
This property is valid for positive values of b, m, and n, and for b≠ 1. For example, log_2 7^4 can be rewritten using this property. log_2 7^4=4 log_2 7
m=b^(log_b(m))
(a^m)^n=a^(m* n)
log_b(b^m)=m
Commutative Property of Multiplication
A logarithm of arbitrary base can be rewritten as the quotient of two logarithms with the same base by using the change of base formula.
log_c a= log_b a/log_b c
This rule is valid for positive values of a,b, and c, where b and c are different than 1.
a= c^x
log_b(a^m)= m* log_b(a)
x= log_c a
.LHS /log_b c.=.RHS /log_b c.
Rearrange equation
A logarithm and a power with the same base undo
each other.
log_b b^x =x and b^(log_b x)=x
In particular, the above equations also hold true for common and natural logarithms.
rcr log 10^x=x & and & 10^(log x)=x [0.8em] ln e^x=x & and& e^(ln x)=x
The general equations will be proved one at a time.
log_b(a^m)= m* log_b(a)
log_b(b) = 1
Identity Property of Multiplication