## Rajasthan Board RBSE Class 11 Maths Chapter 9 Logarithms Ex 9.1

(Q. 1 to 6) Write the following in logarithm form :

Question 1.

2^{6} = 64

Solution:

Logarithm form of 2^{6} = 64 is

log_{2} 64 = 6

Question 2.

10^{4} = 10000

Solution:

Logarithm form of 10^{4} = 10000 is

log_{10}10000 = 4

Question 3.

2^{10} = 1024

Solution:

Logarithm form of 2^{10}= 1024 is

log_{2} 1024 = 10

Question 4.

5^{-2} = \(\frac { 1 }{ 25 }\)

Solution:

Logarithm form of 5^{-2} = \(\frac { 1 }{ 25 }\) is

log_{5}(\(\frac { 1 }{ 25 }\)) = -2.

Question 5.

10^{-3 }= 0.001

Solution:

Logarithm form of 10^{-3} = 0.001 is

log_{10}0.001 = – 3

Question 6.

4^{3/2} = 8

Solution:

Logarithm form of 4^{3/2} = 8 is

log_{4} 8 = 3/2

(Q. 7 to 12) Write the following in the power form :

Question 7.

log_{5} 25 = 2

Solution:

Exponential (power) form of log_{5} 25 = 2 is 5^{2} = 25

Question 8.

log_{3} 729 = 6

Solution:

Exponential (power) form of log_{3} 729 = 6 is 3^{6} = 729.

Question 9.

log_{10} 0.001 = – 3

Solution:

Exponential (power) form of log_{10} 0.001 = – 3 is

10^{-3} = 0.001

Question 10.

log_{10} 0.1 = – 1

Solution:

Exponential form of log_{10} 0.1 = – 1 is 10^{-1} = 0.1

Question 11.

log_{3}( \(\frac { 1 }{ 27 }\)) = -3

Solution:

Exponential form of log_{3}( \(\frac { 1 }{ 27 }\)) = -3 is 3^{-3} = \(\frac { 1 }{ 27 }\)

Question 12.

log_{√2}4 = 4

Solution:

Exponential (power) form of log_{√2}4 = 4 is (√2)^{4} = 4

Question 13.

If log_{81}x = \(\frac { 3 }{ 2 }\), then find the value of x.

Solution:

log_{81}x = \(\frac { 3 }{ 2 }\)

⇒ x = (81)^{3/2}

= (9²)^{3/2} = (9)^{3} = 729

Hence, the value of x is 729.

Question 14.

If log_{125}P = \(\frac { 1 }{ 6 }\) then find the value of P.

Solution:

log_{125} P = \(\frac { 1 }{ 6 }\)

⇒ P = (125)^{1/6}

⇒ P = [(5)^{3}]^{1/6}

⇒ P = 5^{1/2} = √5

Hence, the value of P is √5.

Question 15.

If log_{4} m = 1.5, then find the value of m.

Solution:

log_{4} m = 1.5

⇒ m = (4)^{1.5}

⇒ m = (2^{2} )^{1.5}

⇒ m = 2^{3} = 8

Hence, the value of m is 8.

Question 16.

Prove that :

log_{4}[log_{2}{log_{2} (log_{3} 81)}] = 0

Solution:

We know that log_{m} m^{n} = n log_{m}m

and log_{m}m = 1

∴ log_{m}(m)_{n} = n x log_{m}m = n x 1 = n

⇒ log_{m}(m)^{n} = n …(i)

L.H.S. = log_{4}[log_{2}{log_{2}(log_{3} 81)}]

= log_{4}[log_{2} {log_{2}(log_{3}3^{4})}] (∵ 81 = 3^{4})

= log_{4}[log_{2} {(log_{2}^{4})} [According to equal (i)]

= log_{4}{log_{2}(log_{2}2^{2})} (∵ 4 = 2^{2})

= log_{4}(log_{2}2) [According to equal (i)]

= log_{4}(1) (∵ log_{m}m = 1)

= 0 = R.H.S.

Hence Proved.

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