In this section, we will learn about logarithms with examples and properties.

Definition of Logarithm 

We consider $a>0, a \ne 1$ and $M>0$,  and assume that

a^x =M.

In this case, we will call $x$ to be the logarithm of $M$ with respect to the base $a$. We write this phenomenon as

x= log_a M

(Read as: “$x$ is the logarithm of $M$ to the base $a$”)

∴ a^x =M  ⇒  x=log_a M

On the other hand, if x=log_a M then we have a^x =M.

To summarise, we can say that

a^x =M if and only if x=log_a M.

We now understand the above definition with examples.

Examples of Logarithm

1).  We know that 2³ =8.

In terms of logarithms, we can express it as

3 = log₂8

∴ 2³ = 8 ⇔ 3 = log₂8

2).  Note that $10^{-1}=\frac{1}{10}=0.1$

That is, 10^-1 = 0.1

According to the logarithms, we have

-1 = log₁₀ 0.1

Thus, 10^-1 = 0.1 ⇔ -1 = log₁₀ 0.1

Remarks of Logarithm

(A) If we do not mention the base, then there is no meaning of the logarithms of a number.

(B) The logarithm of a negative number is imaginary.

(C) log_a a=1.

Proof:  As a¹ =a, the proof follows from the definition of the logarithm.

(D)  log_a 1=0.

Proof:   For any a ≠ 0, we have a⁰ =1. Now applying the definition of logarithms, we obtain the result.

Properties of Logarithm

Logarithm has the following four main properties

a). log_a(MN) = log_aM + log_aN

This is called the product rule of logarithms.

b). log_a(M/N) = log_aM – log_aN

This is called the Quotient Rule of logarithms

c). log_aMⁿ =n log_a M

This is called the Power Rule of logarithms

d). log_a M = log_b M × log_a b.

This is the Base Change Rule of logarithms

Solved Examples

Ex1:  Find log₃27

Note that we have 27=3³.

So by the definition of the logarithm, we have

log₃ 27=3 ans.

Ex2:  Find $\log_2 \sqrt{8}$

We have 8=2³

$\therefore \sqrt{8}=(2^3)^{1/2}=(2)^{3 \times 1/2}=2^{3/2}$

Thus, $\sqrt{8}=2^{3/2}$

Now, $\log_2 \sqrt{8}=\log_2 (2)^{3/2}=3/2 \log_2 2=3/2$ ans.

(by the above power rule of logarithms and log_a a=1)

ALSO READ

• Logarithm Formulas with Proofs
• Common Logarithm and Natural Logarithm

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