# Logarithm Formulas and Their Proofs

Here we learn the general laws of logarithms. The list of logarithm formulas are given at the end of this post.

## Proofs of Logarithm Formulas:

Proof of Product Rule of logarithms

loga(MN)=loga M + loga N

Proof:

Suppose that x= loga M

Thus, by the definition of logarithm

ax=M

Again, suppose that y=loga N  ⇒ ay=N

Note that MN=ax ⋅ ay= ax+y

Taking logarithms with respect to the base a on both sides,

loga(MN)=loga ax+y = x+y $[\because \log_a a^k=k]$

= loga M + loga N (plug-in the values of x and y)

Hence, loga(MN)=loga M + loga N (Proved)

In the same way as above, we can prove the quotient rule of logarithms.

Proof of Quotient Rule of Logarithms

loga(M/N)=loga M – loga N

Proof:

Suppose that x=loga M

Thus, by the definition of logarithm

ax=M

Again, suppose that y=loga N  ⇒ ay=N

Note that M/N = $\dfrac{a^x}{a^y}$ =ax-y

Taking logarithms with respect to the base $a$ on both sides,

loga (M/N) = loga ax-y

= x-y $[\because \log_a a^k=k]$

= loga M – loga N (plug-in the values of x and y)

Hence, loga(M/N)=loga M – loga(Proved)

Proof of Power Rule of logarithms

$\log_a M^n=n \log_a M$

Proof:

Suppose that x=loga Mn

Thus, ax=Mn—-(i)

Again, let y=loga M

This implies that ay=M —-(ii)

Combining (i) and (ii) we get that

$a^x=M^n=(a^y)^n$ [$\because a^y=M$]

⇒ ax=any$Equating the powers of a, we obtain that x=ny Putting the values of x and y, we have loga Mn=n loga M (Proved) Proof of Base Change Rule of logarithms loga M= logb M × loga b Proof: Let x=loga M Thus, ax=M —-(i) Now, assume that y=logb M ⇒ by=M —(ii) Again, assume z=loga b ⇒ az=b —-(iii) Combining (i) and (ii), we have$a^x=b^y=(a^z)^b$[$\because a^z=b$by (iii)] ⇒ ax=abz Equating the powers of a, we obtain x=bz Putting the values of x, b and z, we get that loga M= logb M × loga b (Proved) Corollary: logb a =$\dfrac{1}{\log_a b}$Proof: Take M=a in the above base change formula. We have loga a= logb a × loga b ⇒ 1 = logb a × loga b [loga a=1] Hence, it follows that logb a =$\dfrac{1}{\log_a b}$(Proved) ## Logarithm Formulas List 1. loga 1=0 2. loga a = 1 3.$a^{\log_a M}=M\$

4.  loga(MN)=loga M + loga N

5.  loga(M/N)=loga M – loga N

6.  loga Mn = n loga M

7.  loga M= logb M × loga b

8.  logb a = 1/logab

Common logarithm and Natural Logarithm

## FAQs

### Q: What is the product rule of logarithms?

Answer: The product rule of logarithms is given by loga(MN)=loga M + loga N

### Q: What is the quotient rule of logarithms?

Answer: The quotient rule of logarithms is given by loga(M/N)=loga M – loga N