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Q.15 Simplify: log 2+log +log 1/2 5 us 10- 2 21 |
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Answer» `5^(log_(1/5)(1/2)) = 5^(log_(5^(-1))(2^(-1))` `= 5^(log_(5)(2)) = 2` Now, `log_(sqrt2) (4/(sqrt7+sqrt3)) = log_(2^(1/2)) (4/(sqrt7+sqrt3)**(sqrt7- sqrt3)/(sqrt7- sqrt3)) ` `=2log_2 ((4(sqrt7-sqrt3))/(7-3))` `=log_2 (sqrt7- sqrt3)^2` `=log_2 (10-2sqrt21**(10+2sqrt21)/(10-2sqrt21))` `=log_2 ((100-84)/(10-2sqrt21))` `=log_2 (16/(10-2sqrt21))` `=>log_2 16 + log_2 (1/(10-2sqrt21))` Now, the given equation becomes, `5^(log_(1/5)(1/2))+log_(sqrt2) (4/(sqrt7+sqrt3))+ log_(1/2) (1/(10-2sqrt21))) = 2+log_2 16 + log_2 (1/(10-2sqrt21)) + log_(1/2) (1/(10-2sqrt21))` `2+log_2 16 + log_2 (1/(10-2sqrt21)) - log_2 (1/(10-2sqrt21))` `2+4`...[As `log_2 16 = 4`] `=6.` `:. 5^(log_(1/5)(1/2))+log_(sqrt2) (4/(sqrt7+sqrt3))+ log_(1/2) (1/(10-2sqrt21))) = 6` |
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