To solve a logarithmic equation, first combine the logarithmic terms using rules of logarithms.
Some of the rules in logarithm :
log m + log n = log (m x n)
log m - log n = log (m / n)
log m^{n} = n log m
log _{a} a = 1
log _{a} b = 1/log _{b} a
Find :
Problem 1 :
log_{3 }9
Solution :
= log_{3 }9
= log_{3} 3^{2}
= 2 log_{3} 3
= 2 (1)
= 2
Problem 2 :
log_{2 }32
Solution :
= log_{2 }32
= log_{2} 2^{5}
= 5 log_{2} 2
= 5 (1)
= 5
Problem 3 :
log_{2 }√2
Solution :
= log_{2 }√2
= log_{2} 2^{1/2}
= 1/2 log_{2} 2
= 1/2 (1)
= 1/2
Problem 4 :
log_{4 }√2
Solution :
= log_{4 }√2
Problem 5 :
log_{3 }3√3
Solution :
= log_{3 }3√3
Problem 6 :
log_{6 }1
Solution :
= log_{6 }1
= 0
Problem 7 :
log_{8 }8
Solution :
= log_{8 }8
= 1
Problem 8 :
log_{8 }(1/8)
Solution :
= log_{8 }(1/8)
= log_{8 }8^{-1}
= (-1) log_{8 }8
= -1 × 1
= -1
Problem 9 :
log_{1/8 }(1/8)
Solution :
= log_{1/8 }(1/8)
= 1
Problem 10 :
log_{√2 }(1/√2)
Solution :
= log_{√2 }(1/√2)
= log_{√2 }√2^{-1}
= (-1) log_{√2 }√2
= -1 × 1
= -1
Problem 11 :
log_{2 }(1/√2)
Solution :
= log_{2 }(1/√2)
= log_{2 }2^{-1/2}
= (-1/2) log_{2 }2
= -1/2 × 1
= -1/2
Problem 12 :
log_{8 }(1/√2)
Solution :
= log_{8 }(1/√2)
= log_{8 }(1/2)^{1/2}
= log_{8 }(2)^{-1/2}
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