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**

log_{a}(MN)=log_{a} M + log_{a} N

**Proof:**

Suppose that x= log_{a} M

Thus, by the definition of logarithm

a^{x}=M

Again, suppose that y=log_{a} N ⇒ a^{y}=N

Note that MN=a^{x} ⋅ a^{y}= a^{x+y}

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

log_{a}(MN)=log_{a} a^{x+y} = x+y $[\because \log_a a^k=k]$

= log_{a} M + log_{a} N (plug-in the values of x and y)

Hence, log_{a}(MN)=log_{a} M + log_{a} N **(Proved)**

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

**Proof of Quotient Rule of Logarithms **

log_{a}(M/N)=log_{a} M – log_{a} N

**Proof:**

Suppose that x=log_{a} M

Thus, by the definition of logarithm

a^{x}=M

Again, suppose that y=log_{a} N ⇒ a^{y}=N

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

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

log_{a} (M/N) = log_{a} a^{x-y}

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

= log_{a} M – log_{a} N (plug-in the values of x and y)

Hence, log_{a}(M/N)=log_{a} M – log_{a} N **(Proved)**

**Proof of Power Rule of logarithms **

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

**Proof:**

Suppose that x=log_{a} M^{n}

Thus, a^{x}=M^{n}—-(i)

Again, let y=log_{a} M

This implies that a^{y}=M —-(ii)

Combining (i) and (ii) we get that

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

⇒ a^{x}=a^{ny}$

Equating the powers of a, we obtain that x=ny

Putting the values of x and y, we have

log_{a} M^{n}=n log_{a} M **(Proved) **

**Proof of Base Change Rule of logarithms **

log_{a} M= log_{b} M × log_{a} b

Proof:

Let x=log_{a} M

Thus, a^{x}=M —-(i)

Now, assume that y=log_{b} M

⇒ b^{y}=M —(ii)

Again, assume z=log_{a} b

⇒ a^{z}=b —-(iii)

Combining (i) and (ii), we have

$a^x=b^y=(a^z)^b$ [$\because a^z=b$ by (iii)]

⇒ a^{x}=a^{bz}

Equating the powers of a, we obtain x=bz

Putting the values of x, b and z, we get that

log_{a} M= log_{b} M × log_{a} b **(Proved)**

**Corollary:**

log_{b} a = $\dfrac{1}{\log_a b}$

*Proof:* Take M=a in the above base change formula. We have

log_{a} a= log_{b} a × log_{a} b

⇒ 1 = log_{b} a × log_{a} b [log_{a} a=1]

Hence, it follows that log_{b} a = $\dfrac{1}{\log_a b}$ **(Proved)**

## Logarithm Formulas List

1. log_{a} 1=0

2. log_{a} a = 1

3. $a^{\log_a M}=M$

4. log_{a}(MN)=log_{a} M + log_{a} N

5. log_{a}(M/N)=log_{a} M – log_{a} N

6. log_{a} M^{n} = n log_{a} M

7. log_{a} M= log_{b} M × log_{a} b

8. log_{b} a = 1/log_{a}b

**Also read: **

**Logarithm Formulas with Proofs**

**Common logarithm and Natural Logarithm**

## FAQs

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

Answer: The product rule of logarithms is given by log_{a}(MN)=log_{a} M + log_{a} N

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

Answer: The quotient rule of logarithms is given by log_{a}(M/N)=log_{a} M – log_{a} N