solve: log (base 10)384/5+ log (base10)81/32 + 3log (base 10)5/3+log (base 10)1/9.
please help me....
Answers
Step-by-step explanation:
Given :-
log (base 10)384/5+ log (base10)81/32 + 3log (base 10)5/3+log(base 10)1/9.
To find:-
Solbe the given logarithmic expression ?
Solution:-
Given that
The logarithmic expression is
log (base 10)384/5+ log (base10)81/32 + 3log (base 10)5/3+log(base 10)1/9.
Note that 10 is the base in the given logarithmic expression
So it is a common logarithm.
So It can be written as
log 384/5 + log 81/32 + 3log 5/3 + log 1/9
We know that
log a/b = log a - log b
=> log 384-log 5 +log 81-log32 +3(log 5-log3) + log 1-log 9
=> log (2×2×2×2×2×2×2×3)- log 5 + log (3×3×3×3)- log (2×2×2×2×2) + 3 log 5
- 3 log 3 + log 1 - log(3×3)
=> log (2^7×3) - log 5 + log 3^4-log 2^5
+3 log 5 - 3log 3 +log 1 - log 3^2
We know that
log ab = log a + log b
=>log 2^7 + log 3 - log 5 + log 3^4-log 2^5 +3 log 5 - 3log 3 +log 1 - log 3^2
We know that
log a^m = m log a
=> 7 log 2 + log 3 - log 5 + 4 log 3 -5 log 2 + 3 log 5 - 3 log 3 + log 1 - 2 log 3
=> log 1 + (7log 2 -5 log 2) +(log 3 +4log 3 - 3log3 -2 log 3) + (-log 5 +3 log 5)
=> log 1 + 2 log 2 + 0 log 3 +2 log 5
we know that
log 1 = 0
=> 0+2 log 2 +0 + 2 log 5
=> 2 log 2 + 2 log 5
=> 2( log 2+ log 5)
We know that
log ab = log a + log b
=> 2(log 2×5)
=> 2 log 10
It can be written as
=> 2 log (base 10) 10
We know that
log (base a ) a = 1
=> 2× 1
=> 2
Answer:-
The value of the given logarithmic expression is 2
Used formulae:-
- log ab = log a + log b
- log a/b = log a - log b
- log a^m = m log a
- log (base a ) a = 1
- log (base 10) 1 = 0