Apply the properties of logarithms to simplify the expressions: (Criteria A)a) log39+log327b) log28−log22c) 3log105−log1025d) lnx+lnye) logaan
Question
Apply the properties of logarithms to simplify the expressions:
(Criteria A)
a) log_3 9 + log_3 27
b) log_2 8 - log_2 2
c) 3 log_{10} 5 - log_{10} 25
d) ln x + ln y
e) log_a a^n
Solution
Sure, here are the simplified expressions using the properties of logarithms:
a) log39 + log327 Using the property loga + logb = log(ab), we get: log(39*327) = log12753
b) log28 - log22 Using the property loga - logb = log(a/b), we get: log(28/22) = log1.2727
c) 3log105 - log1025 First, use the property n*loga = log(a^n) on 3log105 to get log(105^3). Then, use the subtraction property: log(105^3/1025) = log(1157625/1025) = log1129.878
d) lnx + lny Using the property ln(a) + ln(b) = ln(ab), we get: ln(xy)
e) loga^an Using the property n*loga = log(a^n), we get: log(a^an) = log(a^(a^n))
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