Question

Wyznacz wartośc parametru a tak, aby podana obok równanialiczba była rozwiązaniem tego równania:

Oblicz wartośc wyrażenia:

Wiedząc, że log12=a i log2=b, oblicz log90

Answer 1

z.1

( 2 + a)^2 - (2 - a)^2 = 40

4 + 4 a + a^2 - 4 + 4 a -a^2 = 40

8a = 40

a = 5

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( 2*(-1) - a) *(2 *(-1) + a) - ( -1 - a)^2 = 2

( -2 - a)*( -2 + a) - (-2 + a)^2 = 2

4 - a^2 - ( 4 - 4 a + a^2 ) = 2

-2 a^2 + 4 a - 2 = 0  / : (-2)

a^2 - 2 a + 1 = 0

( a - 1)^2 = 0

a = 1

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( 4 - a)*( 4 + a) - ( 4 - a)^2 = 8

16 - a^2 - (16 - 8 a + a^2 = 8

- 2 a^2 + 8 a - 8 = 0  / : ( -2)

a^2 - 4 a + 4 = 0

( a - 2)^2 = 0

a = 2

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z.2

[ log p(128) + log 32^(1/3) ] / log (2 p(2) ] =

= [ log p(64 *2) + log ( 2^5)^(1/3) ] / log 2*2^(1/2)  =

= [  log 8 p(2) + log 2^(5/3) ] / log 2^(3/2)  =

= [ log 2^3 * 2^(1/2) + log 2 ^(5/3) ] / log 2 ^(3/2) =

= [ log 2 ^(7/2)  + log 2^(5/3) ] / log 2^(3/2) =

= [ log 2^(21/6) + log 2^(10/6) ] / log 2 ^(3/2) =

= log [ 2^ (21/6) * 2^(10/6) ] / log 2 ^(3/2) =

= log 2 ^(31/6) / log 2^(3/2) =

[ (31/6) * log 2 ] / [ (3/2)* log 2 ] =

= (31/6) *(2/3) = 62/18 = 31/9

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z.3

log 12 = a

log  2 = b

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log 90 = log [ 180 / 2 ] = log 180 - log 2 = log [15*12] - log 2 =

=  log 15 + log 12 - log 2  = log [ 1,5 *10] + log 12 - log 2 =

= log 1,5 + log 10 + log 12 - log 2 = log [12/8] + 1 + log 12 - log 2 =

= log 12 - log 8 + 1 + log 12 - log 2 = 2 log 12 - log 2 - log 2^3  + 1 =

= 2 log 12 - log 2 - 3 log 2 + 1 = 2 log 12 - 4 log 2 + 1 =

= 2*a - 4*b + 1

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