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 $a^x=b$ can be written in the logarithmic language as follows: $x=\log_a b.$ Moreover, \[a^x=b \text{ if and only if } x=\log_a b.\] 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.
Table of Contents
Logarithm Formulas (Log Rules)
• Product Rule of Logarithms:
$\log_a(MN)=\log_a M+\log_a N$
• Quotient Rule of Logarithms:
$\log_a(M/N)=\log_a M-\log_a N$
• Power Rule of Logarithms:
$\log_a M^n=n\log_a M$
• Base Change Rule of Logarithms:
$\log_a M=\log_b M \times \log_a b$
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
| \[\log_a(MN)=\log_a M+\log_a N\] |
Proof: Let $\log_a M=x$ and $\log_a N=y$
So by the definition of the logarithm, we have
$M=a^x$ and $N=a^y$
⇒ $MN=a^x . a^y$
⇒ $MN=a^{x+y}$
Taking logarithm with base $a$, we get
$\log_a (MN)=\log_a a^{x+y}$
⇒ $\log_a (MN)=x+y$ $[\because \log_a a^n=n]$
⇒ $\log_a (MN)=\log_a M+\log_a N$
[since $x=\log_a M$ and $y=\log_a N$]
∴ the product rule of logarithms is proved. ♣
Proof of quotient rule of logarithms
| \[\log_a(M/N)=\log_a M-\log_a N\] |
Proof: Let $\log_a M=x$ and $\log_a N=y$
So by the definition of the logarithm, we have
$M=a^x$ and $N=a^y$
⇒ $M/N=a^x /a^y$
⇒ $M/N=a^{x-y}$
Taking logarithm with base $a$, we get
$\log_a (M/N)=\log_a a^{x-y}$
⇒ $\log_a (M/N)=x-y$ $[\because \log_a a^n=n]$
⇒ $\log_a (M/N)=\log_a M-\log_a N$
[since $x=\log_a M$ and $y=\log_a N$]
∴ the quotient rule of logarithms is proved. ♣
Proof of power rule of logarithms
| \[\log_a M^n=n\log_a M\] |
Proof: Let $\log_a M^n=x$ and $\log_a M=y$
∴ To prove the result, we need to show that $x=ny$
Now by the definition of the logarithm,
$M^n=a^x$ and $M=a^y$
⇒ $(a^y)^n=a^x$
⇒ $a^{ny}=a^x$
Comparing the powers of $a$ on both sides, we obtain that
$x=ny.$ In other words,
$\log_a M^n=n\log_a M$
∴ the power rule of logarithms is proved. ♣
As a corollary, we can prove the following:
Corollary:
| \[\log_a \sqrt[n]{M}=\frac{1}{n} \log_a M\] |
Proof: Note that $\sqrt[n]{M}=M^{1/n}$
So by the power rule of logarithms, we have
$\log_a \sqrt[n]{M}=\log_a M^{1/n}=\frac{1}{n} \log_a M$ ♣
Proof of base change rule of logarithms
| \[\log_a M=\log_b M \times \log_a b\] |
Proof: Let $\log_a M=x,$ $\log_b M=y$ and $\log_a b=w$
∴ To prove the result, we need to establish that $x=yw$
By the definition of the logarithm, one has
$M=a^x,$ $M=b^y$ and $b=a^w$
⇒ $a^x=b^y$ and $b=a^w$
⇒ $a^x=(a^w)^y$
⇒ $a^x=a^{yw}$
Comparing the powers, $x=yw$
∴ $\log_a M=\log_b M \times \log_a b$
So the base change rule of logarithms is proved. ♣
As a corollary, we prove the following:
Corollary:
| The reciprocal of $\log_a b$ is $\log_b a.$ In other words, $\frac{1}{\log_a b}=\log_b a$ |
Proof: By the base change rule of logarithm, we have
$\log_a b \times \log_b a=\log_a a=1$
⇒ $\frac{1}{\log_a b}=\log_b a$ ♣
Complete List of Logarithm Formulas:
1. $x=\log_a b \iff a^x=b$
2. $\log_a a=1$
3. $a^{\log_a M}=M$
4. $\log_a a^n=n$
5. $\log_a M^n=n \log_a M$
6. $\log_a (MN)=\log_a M + \log_a N$
7. $\log_a (M/N)=\log_a M – \log_a N$
8. $\log_a \sqrt{M}=\frac{1}{2}\log_a M$
9. $\log_a \sqrt[n]{M}=\frac{1}{n}\log_a M$
10. $\frac{1}{\log_a b}=\log_b a$
Application of Logarithm Formulas:
As an application of the above logarithm rules, we can learn
• how to add logarithms: For example, lets add $\log_2 3$ and $\log_2 7.$ Note that $\log_2 3+\log_2 7$ $=\log_2 (3 . 7)$ $=\log_2 21$, by the product rule.
• how to subtract logarithms: For example, lets subtract $\log_2 3$ from $\log_2 9.$ Note that $\log_2 9-\log_2 3$ $=\log_2 (9/3)$ $=\log_2 3$, by the quotient rule.
• how to multiply logarithms: For example, lets multiply $\log_2 3$ with $\log_3 7.$ Note that $\log_2 3\times \log_3 7$ $=\log_2 7$, by the base change rule.
• how to divide logarithms: For example, lets divide $\log_2 3$ by $\log_7 3.$ Note that
$\log_2 3 \div \log_7 3$
$=\log_2 3 \times \frac{1}{\log_7 3}$
$=\log_2 3 \times \log_3 7$ $[\because \frac{1}{\log_b a}=\log_a b]$
$=\log_2 7$, by the base change rule.
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