Question

What is logarithm? How to do log? Explain in about 1-2 page with 2-3 examples.

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Answer 1

▫Definition of logarithm :

➡ If
${a}^{x} = b$
with a > 0, b > 0, a ≠ 1, then x is called the logarithm of the number b to the base a.

It is denoted as
$x = log_{a}b$
and is read as "x is the logarithm of b to the base a".

▫Rules :

There are four basic rules to do logarithms.

$1. \: log_{a}(bc) = log_{a}b + log_{a}c \\ \\ 2. \: log_{a}( \frac{b}{c } ) = log_{a}b - log_{a}c \\ \\ 3. \: log_{a} {b}^{n} = n \: log_{a}b \\ \\ 4. \: log_{a}b = log_{p}b \times log_{a}p$
where a, b, c, p > 0, a ≠ 1, b ≠ 1 and n is any real number.

After these basic formulae, it will be easy to understand the following ones.

$1. \: log_{a}1 = 0 \\ \\ 2. \: log_{a}a = 0 \\ \\ 3. \: {a}^{ log_{a}b} = b \\ \\ 4. \: log_{b} a\times log_{a}b = 1 \\ \\ 5. \: log_{b}a = \frac{1}{ log_{a}b } \\ \\ 6. \: log_{b}p = \frac{ log_{a}p }{ log_{a}b }$

Depending on these formulae, let us solve a few problems.

▫Examples :

1. Prove that :
$log_{2}10 - log_{ 8 }125 = 1$

Proof.

Now,

$log_{8}125 \\ = log_{8} {5}^{3} \\ = 3. \frac{1}{ log_{5}8} \\ = 3. \frac{1}{ log_{5} {2}^{3} } \\ = 3. \frac{1}{3 log_{5}2 } \\ = log_{2}5$

Then, L.H.S.
$= log_{2}10 - log_{2}125 \\ = log_{2}10 - log_{2}5 \\ = log_{10}5 \\ = log_{2}2 \\ = 1$
= R.H.S. [Proved]

2. Prove that :
$log_{4}2 \times log_{2}3 = log_{5}3 \times log_{4}5$

Proof.

L.H.S.
$= log_{4}2 \times log_{2}3 \\ = log_{4}3 \\ = log_{5}3 \times log_{4}5$
= R.H.S. [Proved]

3. If base is 0.01, find the logarithm of 0.000001.

Ans.

Let us consider, the required logarithmic value = x.

Then,
$log_{0.01}0.000001 = x \\ or \: \: {(0.01)}^{x} = 0.000001 = {(0.01)}^{3} \\ \\ so \: \: x = 3 \\ \\ hence \\ log_{0.01}0.000001 = 3$
(Ans.)

Thank you for your question.

Answer 2

hy
here is your answer brother
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What is logarithm?

A logarithm is the power to which a number must be raised in order to get some other ... 23 = 8. In general, you write log followed by the base number as a subscript.
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How to do log?

1) Multiplication inside the log can be turned into addition outside the log, and vice versa.

2) division inside the log can be turned into subtraction outside the log, and vice versa.

3)An exponent on everything inside a log can be moved out front as multiplier and vice versa.

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example 1)  Find the logarithms of: 

(i) 1728 to the base 2√3

let x denoted the required lagorithm

therefore log2√3 1728=x

or, (2√3)x = 1728 = 26 ∙ 33 = 26 ∙ (√3)6 

or, (2√3)x = (2√3)6 

Therefore, x = 6. 

example2). Proof that, log2 log2 log2 16 = 1. 

L. H. S. = log2 log2 log2 24

  = log2 log2 4 log2 2 

  = log2 log2 22   [since log2 2 = 1] 

  = log2 2 log2 2 

  = 1 ∙ 1 

  = 1. Proved.

example3) :- a) log 2 4 + log 2 5

b) log a 28 – log a 4

c) 2 log a 5 – 3 log a 2

solution:-a) log 2 4 + log 2 5 = log 2 (4 × 5) = log 2 20

b) log a 28 – log a 4 = log a (28 ÷ 4) = log a 7

c) 2 log a 5 – 3 log a 2 = log a 52 – log a 23 = log a

example4):-Evaluate 2 log3 5 + log3 40 – 3 log3 10

2 log3 5 + log3 40 – 3 log3 10 
= log3 52 + log3 40 – log3 103 
= log3 25 + log3 40 – log3 1000

= log3 
= log3 1 
= 0