Reference

Properties of Logarithms

Rule

Product Property of Logarithms

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

Proof

Start by recalling the definition of a logarithm. log_b a=c ⇔ a=b^c Before proving the desired property, two other identities will be justified. The first equation of the definition states that c=log_b a. Therefore log_b a can be substituted for c in the second equation. a=b^c Substitute a=b^(log_b a) Furthermore, the second equation states that a is equal to b^c. This means that b^c can be substituted for a in the first equation. log_b a=c Substitute log_b b^c=c The obtained identities will be used together with the Product of Powers Property to prove the Product Property of Logarithms.
log_b mn
log_b(m* n)

m=b^(log_b(m))

log_b(b^(log_b m)* b^(log_b n))
log_b(b^(log_b m+log_b n))

log_b(b^m)=m

log_b m+log_b n


Rule

Quotient Property of Logarithms

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

Proof


Start by recalling the definition of a logarithm. log_b a=c ⇔ a=b^c Before proving the desired property, two other identities will be justified. The first equation of the definition states that c=log_b a. Therefore log_b a can be substituted for c in the second equation. a=b^c Substitute a=b^(log_b a) Furthermore, the second equation states that a is equal to b^c. This means that b^c can be substituted for a in the first equation. log_b a=c Substitute log_b b^c=c The obtained identities will be used together with the Quotient of Powers Property to prove the Quotient Property of Logarithms.
log_b m/n

m=b^(log_b(m))

log_b(b^(log_b m)/b^(log_b n))
log_b(b^(log_b m-log_b n))

log_b(b^m)=m

log_b m-log_b n
Rule

Power Property of Logarithms

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

Proof

Start by recalling the definition of a logarithm. log_b a=c ⇔ a=b^c Before proving the desired property, two other identities will be justified. The first equation of the definition states that c=log_b a. Therefore log_b a can be substituted for c in the second equation. a=b^c Substitute a=b^(log_b a) Furthermore, the second equation states that a is equal to b^c. This means that b^c can be substituted for a in the first equation. log_b a=c Substitute log_b b^c=c The obtained identities will be used together with the Power of a Power Property to prove the Power Property of Logarithms.
log_b m^n

m=b^(log_b(m))

log_b (b^(log_b m) )^n
log_b b^((log_b m )* n)

log_b(b^m)=m

(log_b m)* n
nlog_b m
Rule

Change of Base Formula

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.

Proof

Let x=log_c a. Therefore, by the definition of a logarithm, it is known that a=c^x. x=log_c a ⇔ a=c^x By the Reflexive Property of Equality, log_b a is equal to itself.
log_b a=log_b a
log_b a=log_b c^x

log_b(a^m)= m* log_b(a)

log_b a=xlog_b c
log_b a= log_c alog_b c
log_b a/log_b c=log_c a
log_c a=log_b a/log_b c ✓
Rule

Inverse Properties of Logarithms

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

Proof

The general equations will be proved one at a time.

log_b b^x =x

This identity can be proved by using the Power Property of Logarithms and the definition of a logarithm.
log_b b^x

log_b(a^m)= m* log_b(a)

x log_b b

log_b(b) = 1

x(1)
x ✓
The logarithm of b^x with base b is equal to x.

b^(log_b x)=x

Let log_b x=a. Therefore, by the definition of a logarithm, b^a=x. log_b x=a ⇔ b^a=x This will be used to prove the identity.
b^(log_b x)
b^a
x ✓
Therefore, b to the power of log_bx is equal to x.
Exercises