Question

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Answer 1

Question 38 :  

If n ∈ N, prove that,  1/log₂x + 1/log₃x + 1/log₄x + ........ 1/logₙx = 1/log_{ n! }

Answer :  

Refer to attachment

 

Question 38 :

If a² + b² = 7ab, prove that  

log[ 1/3 ( a + b ) ] = 1/2 [ log a + log b ]  

Answer :  

Given :  

a² + b² = 7ab  

Adding 2ab on both sides  

⇒ a² + b² + 2ab = 7ab + 2ab  

⇒ ( a + b )² = 9ab  

⇒ ( a + b )² / 9 = ab  

⇒ ( a + b )² / 3² = ab  

⇒ [ ( a + b ) / 3 ]² = ab  

Taking log on both sides  

⇒ log[ 1/3( a + b ) ]² = log ab  

⇒ 2log[ 1/3( a + b ) ] = log a + log b  

⇒ log[ 1/3( a + b ) ] = 1/2( log a + log b )  

Hence proved.  

Question 40 :  

If a, b, c are in GP, prove that  

(i) logₓa, logₓ b, logₓ c are in AP  

(ii) log_a x, log_b x, log_c x are in HP  

Answer :  

Refer to attachment  

Question 41 :  

If log₃ 2, log₃ ( 2^x - 5 ), log₃ ( 2^x - 7/2 ) are in AP, find the value of x.  

Answer :  

Refer to attachment  

Question 42 :  

Prove that :  

(i) log_a n / log_(ab) n = 1 + log_a b

(ii) [ log_a x. log_b x ] / [ log_a x + log_b x ]  = log_(ab) x

Answer :  

Refer to attachment  

Question :

Find the value of x in each of the following :

(i) log_( √8 ) x = 10/3  

⇒ ( √8 )^( 10/3 ) = x  

⇒ ( 2³ )^( 5/3 ) = x  

⇒ 2⁵ = x  

⇒32 = x  

(ii) log_( √8 ) x = 3^( 1/3 )  

⇒( √8 )^( ∛3 ) = x  

⇒ x ≈ 4.4796  

(iii) ( log_e 2 )( logₓ 625 ) = ( log₁₀ 16 )( log_e 10 )  

Since log_a b = log b / log a  

⇒ ( log 2 / log e )( log 5^4 / log x ) = ( log 2^4 / log 10 )( log 10 / log e )  

⇒ ( log 2 )( 4log 5 ) = ( 4log 2 )( log x )  

⇒ log 5 = log x  

⇒ x = 5

Answer 2

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