Question

1. Oblicz:

a)log₂8 + log½4 + log₉81

b)log 0,1 + log 1000 + log0,1 10

c)log₃27 - log₄2 - log₂ 1/8

d)log₃₆6 + log₁₆4 - log₈2

2. Oblicz:

a)log₆2 + log₆18

b)log 15 - log 3/2

c)log₆4 +log₆27 - log₆3

d)log 1/2 - log 25 - log 2

Answer 1

1. a)log₂8 + log½4 + log₉81

    3+(-2)+2=3

b)log 0,1 + log 1000 + log0,1 10

    -1+3+(-1)=-2+3=1

c)log₃27 - log₄2 - log₂ 1/8

     3-1/2-(-3)=5 1/2

d)log₃₆6 + log₁₆4 - log₈2

   1/2+1/2-1/3=2/3

 

2. Z własności logarytmów o tych samych podstawach (np. log₆ a + log ₆b = log ₆(a*b) )

a)log₆2 + log₆18

    log₆(2*18)=log₆36=2

b)log 15 - log 3/2

     log (15:3/2)= log 10=1

c)log₆4 +log₆27 - log₆3

    log₆(4*27)-log₆3

     log₆108-log₆3

     log₆(108:3)=log₆36=2

d)log 1/2 - log 25 - log 2

     log(1/2:25)-log2

      log 1/50 - log2

      log(1/50 : 2)=log 1/100=-2

Answer 2

1.

logg₂8 + log½4 + log₉81 = log₂ 2³ + log½ (1/2)^-2 + log₉ 9² = 3 + (-2) + 2 = 3 

 

b)

log 0,1 + log 1000 + log0,1 10 = log 10^(-1) + log 10³ + log0,1 0,1^(-1) =  -1 + 3 + (-1) = 1

 

c)

log₃27 - log₄2 - log₂ 1/8 = log₃ 3³ - log₄ 4^(1/2) - log₂ 2^(-3) =   3 - 1/2 - (-3) = 2,5 + 3 = 5,5

 

d)

log₃₆6 + log₁₆4 - log₈2 = log₃₆ 36^(1/2) + log₁₆ 16^(1/2) - log₈ 8^(1/3) = 1/2 + 1/2 - 1/3 = 2/3

 

2.

a)

log₆2 + log₆18 = log₆ (2 * 18) = log₆ 36 log₆ 6² = 2

 

b)

log 15 - log 3/2 = log (15 : 3/2) = log (15 * 2/3) = log 10 = 1

 

c)

log₆4 +log₆27 - log₆3 =log₆(4 * 27) - log₆3 =log₆108 - log₆3 =log₆(108 : 3)=log₆ 36= log₆ 6² =2

 

d)

log 1/2 - log 25 - log 2 = log(1/2 : 25) - log 2  = log(1/2 * 1/25) - log 2 = log (1/50 : 2) = log (1/50 * 1/2) = log 1/100 = log 10^(-2) = -2