Interview
Logarithms
- Course
- Institution
Definition of logarithms, properties of all logarithms, product property, ono to one, change of base property, Quotient Property, Power Property
[Show more]
Seller
Follow
dinuarya
Content preview
Logarithms and Its Basic PropertiesDefinition of Logarithm: If 𝑦 and 𝑎 are positive real numbers(𝑎 ≠ 1), then 𝑥 = 𝑙𝑜𝑔 𝑦 if
and only if 𝑎 = 𝑦.
The notation 𝑙𝑜𝑔 𝑦 is read as “log to the base 𝑎 of 𝑦 ”. In the equation 𝑥 = 𝑙𝑜𝑔 𝑦 , 𝑥 is
known as the logarithm, 𝑎 is the base and 𝑦 is the argument.
Note: 1. The above definition indicates that a logarithm is an exponent.
2. Logarithm of a number may be negative but the argument of logarithm must be positive.
The base must also be positive and not equal to 1.
3. Logarithmic Form Exponential Form
𝑥 = 𝑙𝑜𝑔 𝑦 𝑎 =𝑦
4. Logarithm of zero doesn’t exist.
5. Logarithms of negative real numbers are not defined in the system of real numbers.
6. Log to the base “10” is called Common Logarithm and Log to the base “𝑒” is called Natural
Logarithm.(𝑒 = 2.7182818284 … )
7. If base of logarithm is not given, we’ll consider it Natural Logarithm.
Some examples of logarithmic form and their corresponding exponential form:
S. No. Logarithmic form Exponential form
1 5 = 𝑙𝑜𝑔 32 2 = 32
2 4 = 𝑙𝑜𝑔 81 3 = 81
3 3 = 𝑙𝑜𝑔 125 5 = 125
4 4 = 𝑙𝑜𝑔 10000 10 = 10000
5 1 1
−2 = 𝑙𝑜𝑔 7 =
49 49
6 0 = 𝑙𝑜𝑔 1 𝑒 =1
,Why do we study logarithms: Sometimes multiplication, subtraction and exponentiation
become so lengthy and tedious to solve. Logarithms covert the problems of multiplication
into addition, division into subtraction and exponentiation into multiplication, which are easy
to solve.
Properties of Logarithms: If 𝑎, 𝑏 and 𝑐 are positive real numbers, 𝑎 ≠ 1 and 𝑛 is any real
number, then
1. Product property: 𝒍𝒐𝒈𝒂 (𝒃. 𝒄) = 𝒍𝒐𝒈𝒂 𝒃 + 𝒍𝒐𝒈𝒂 𝒄
For example: 𝑙𝑜𝑔 (187) = 𝑙𝑜𝑔 (11 × 17) = 𝑙𝑜𝑔 11 + 𝑙𝑜𝑔 17
𝒃
2. Quotient property: 𝒍𝒐𝒈𝒂 = 𝒍𝒐𝒈𝒂 𝒃 − 𝒍𝒐𝒈𝒂 𝒄
𝒄
For example: 𝑙𝑜𝑔 = 𝑙𝑜𝑔 51 − 𝑙𝑜𝑔 7
3. Power property: 𝒍𝒐𝒈𝒂 𝒃𝒏 = 𝒏. 𝒍𝒐𝒈𝒂 𝒃
For example: 𝑙𝑜𝑔 (10000) = 𝑙𝑜𝑔 (10 ) = 4. 𝑙𝑜𝑔 10
4. One to One property: 𝒍𝒐𝒈𝒂 𝒃 = 𝒍𝒐𝒈𝒂 𝒄 if and only if 𝒃 = 𝒄.
For example: If 𝑙𝑜𝑔 (𝑎) = 𝑙𝑜𝑔 (15) then 𝑎 = 15.
5. 𝒍𝒐𝒈𝒂 𝟏 = 𝟎
For example: 𝑙𝑜𝑔 (1) = 0 , 𝑙𝑜𝑔 (1) = 0 , 𝑙𝑜𝑔 (1) = 0 etc.
6. 𝒍𝒐𝒈𝒂 𝒂 = 𝟏
For example: 𝑙𝑜𝑔 (10) = 1 , 𝑙𝑜𝑔 (𝑒) = 1 etc.
7. 𝒍𝒐𝒈𝒂 𝒂𝒏 = 𝒏
For example: 𝑙𝑜𝑔 10 = 4
8. 𝒂𝒍𝒐𝒈𝒂 (𝒏) = 𝒏, where 𝒏 > 0
( )
For example: 2 =8
𝒍𝒐𝒈(𝒃) 𝒍𝒐𝒈𝒄 (𝒃)
9. Change of base property: 𝒍𝒐𝒈𝒂 𝒃 = = provided that 𝒄 ≠ 𝟏.
𝒍𝒐𝒈(𝒂) 𝒍𝒐𝒈𝒄 (𝒂)
( )
For example: 𝑙𝑜𝑔 (3) = ( )
(Here we changed the base to 10)
Some solved problems:
Example 1. Convert the following exponential forms into logarithmic forms:
(i) 9 = 729 (ii) 7 = 16807 (iii) 2 = 1024
(iv) 10 = 0.001 (v) 4 = 0.0625 (vi) 5 = 0.0016
(vii) 10 = 1 (viii) 8 = 1
Ans. (i) Given that 9 = 729
⇒ 𝑙𝑜𝑔 729 = 3 (𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛)
which is required logarithmic form.
, OR
Given that 9 = 729
Taking logarithm on both sides, we get
𝑙𝑜𝑔9 = 𝑙𝑜𝑔729
⇒ 3 𝑙𝑜𝑔9 = 𝑙𝑜𝑔729 (𝑢𝑠𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 )
⇒ 3=
( )
⇒ 3 = 𝑙𝑜𝑔 729 𝑢𝑠𝑒𝑑 𝑙𝑜𝑔 𝑏 = ( )
which is required logarithmic form.
(ii) Given that 7 = 16807
⇒ 𝑙𝑜𝑔 16807 = 5 (𝑏𝑦 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑖𝑜𝑛)
which is required logarithmic form.
OR
Given that 7 = 16807
Taking logarithm on both sides, we get
𝑙𝑜𝑔7 = 𝑙𝑜𝑔16807
⇒ 5 𝑙𝑜𝑔7 = 𝑙𝑜𝑔16807 (𝑢𝑠𝑒𝑑 𝑝𝑜𝑤𝑒𝑟 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 )
⇒ 5=
( )
⇒ 5 = 𝑙𝑜𝑔 16807 𝑢𝑠𝑒𝑑 𝑙𝑜𝑔 𝑏 = ( )
which is required logarithmic form.
(iii) Given that 2 = 1024
⇒ 𝑙𝑜𝑔 1024 = 10
which is required logarithmic form.
(iv) Given that 10 = 0.001
⇒ 𝑙𝑜𝑔 0.001 = −3
which is required logarithmic form.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller dinuarya. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75323 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now