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
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.
Table of Contents
Logarithm Formulas (Log Rules)
• Product Rule of Logarithms:
• Quotient Rule of Logarithms:
• Power Rule of Logarithms:
• Base Change Rule of Logarithms:
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
Proof: Let
So by the definition of the logarithm, we have
⇒
⇒
Taking logarithm with base
⇒
⇒
[since
∴ the product rule of logarithms is proved. ♣
Proof of quotient rule of logarithms
Proof: Let
So by the definition of the logarithm, we have
⇒
⇒
Taking logarithm with base
⇒
⇒
[since
∴ the quotient rule of logarithms is proved. ♣
Proof of power rule of logarithms
Proof: Let
∴ To prove the result, we need to show that
Now by the definition of the logarithm,
⇒
⇒
Comparing the powers of
∴ the power rule of logarithms is proved. ♣
As a corollary, we can prove the following:
Corollary:
Proof: Note that
So by the power rule of logarithms, we have
Proof of base change rule of logarithms
Proof: Let
∴ To prove the result, we need to establish that
By the definition of the logarithm, one has
⇒
⇒
⇒
Comparing the powers,
∴
So the base change rule of logarithms is proved. ♣
As a corollary, we prove the following:
Corollary:
The reciprocal of |
Proof: By the base change rule of logarithm, we have
⇒
Complete List of Logarithm Formulas:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Application of Logarithm Formulas:
As an application of the above logarithm rules, we can learn
• how to add logarithms: For example, lets add
• how to subtract logarithms: For example, lets subtract
• how to multiply logarithms: For example, lets multiply
• how to divide logarithms: For example, lets divide
Related Topics |
This article is written by Dr. T, an expert in Mathematics (PhD). On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.