Logarithm Rules | Logarithm Formulas

For the basic concepts of the logarithm, we refer to our page “an introduction to logarithm”. To express the power of a number, we use the concept of the logarithm. Note that ax=b can be written in the logarithmic language as follows: x=logab. Moreover, ax=b if and only if x=logab.  In this section, we will discuss the fundamental laws of logarithms. One can deduce many more nice formulas from these formulas.

 

A complete list of logarithm formulas/rules is provided at the end of the discussion. The following are the main logarithm formulas and we give their proofs here.

Logarithm Formulas (Log Rules)

Product Rule of Logarithms: 

loga(MN)=logaM+logaN

Quotient Rule of Logarithms:

loga(M/N)=logaMlogaN

Power Rule of Logarithms:

logaMn=nlogaM

Base Change Rule of Logarithms:

logaM=logbM×logab

Using the above formulas, we can do many things. For example, we can learn how to add, subtract, divide and multiply logarithms. Before providing such examples, let us first learn how to prove the above logarithm formulas.

We first prove a crucial logarithm formula.

Theorem: Prove that loga an = n.

Proof: Let x = loga an…(i)

To prove the result, it is enough to show that x=n.

From (i), we have that

ax = an.

Comparing the powers, we get that x=n.

In other words, n = loga an ♣

 

Proof of all Logarithm Formulas

Proof of product rule of logarithms

loga(MN)=logaM+logaN

Proof: Let logaM=x and logaN=y

So by the definition of the logarithm, we have

M=ax and N=ay

MN=ax.ay

MN=ax+y

Taking logarithm with base a, we get

loga(MN)=logaax+y

loga(MN)=x+y [logaan=n]

loga(MN)=logaM+logaN

[since x=logaM and y=logaN]

∴ the product rule of logarithms is proved. ♣

 

Proof of quotient rule of logarithms

loga(M/N)=logaMlogaN

Proof: Let logaM=x and logaN=y

So by the definition of the logarithm, we have

M=ax and N=ay

M/N=ax/ay

M/N=axy

Taking logarithm with base a, we get

loga(M/N)=logaaxy

loga(M/N)=xy [logaan=n]

loga(M/N)=logaMlogaN

[since x=logaM and y=logaN]

∴ the quotient rule of logarithms is proved. ♣

 

Proof of power rule of logarithms

logaMn=nlogaM

Proof: Let logaMn=x and logaM=y

∴ To prove the result, we need to show that x=ny

Now by the definition of the logarithm,

Mn=ax and M=ay

(ay)n=ax

any=ax

Comparing the powers of a on both sides, we obtain that

x=ny. In other words,

logaMn=nlogaM

∴ the power rule of logarithms is proved. ♣

 

As a corollary, we can prove the following:

Corollary: 

logaMn=1nlogaM

Proof: Note that Mn=M1/n

So by the power rule of logarithms, we have

logaMn=logaM1/n=1nlogaM

 

Proof of base change rule of logarithms

logaM=logbM×logab

Proof: Let logaM=x, logbM=y and logab=w

∴ To prove the result, we need to establish that x=yw

By the definition of the logarithm, one has

M=ax, M=by and b=aw

ax=by and b=aw

ax=(aw)y

ax=ayw

Comparing the powers, x=yw

logaM=logbM×logab

So the base change rule of logarithms is proved. ♣

 

As a corollary, we prove the following:

Corollary:

The reciprocal of logab is logba. In other words, 1logab=logba

Proof: By the base change rule of logarithm, we have

logab×logba=logaa=1

1logab=logba

 

Complete List of Logarithm Formulas:

1.  x=logabax=b

2.  logaa=1

3.  alogaM=M

4.  logaan=n

5.  logaMn=nlogaM

6.  loga(MN)=logaM+logaN

7.  loga(M/N)=logaMlogaN

8.  logaM=12logaM

9.  logaMn=1nlogaM

10.  1logab=logba

 

Application of Logarithm Formulas:

As an application of the above logarithm rules, we can learn

how to add logarithms: For example, lets add log23 and log27. Note that log23+log27 =log2(3.7) =log221, by the product rule.

how to subtract logarithms: For example, lets subtract log23 from log29. Note that log29log23 =log2(9/3) =log23, by the quotient rule.

how to multiply logarithms: For example, lets multiply log23 with log37. Note that log23×log37 =log27, by the base change rule.

how to divide logarithms: For example, lets divide log23 by log73. Note that

log23÷log73

=log23×1log73

=log23×log37 [1logba=logab]

=log27, by the base change rule.

 

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