1)Find all x for which the inequalities: log(x+2)*(x+4) to the base 3+ log(x+2)to the base 1/3 is less than 1/2log7 to the base sq.rt.of 3 is valid? 2) If a=log 18 to the base 12 and log 54 to the base 24, then ab+5(a-b)=?

Asked by  | 15th Apr, 2013, 06:26: PM

Expert Answer:

1.  log(x+2)*(x+4) to the base 3+ log(x+2)to the base 1/3 < 1/2log7 to the base sq.rt.of 3 
[log(x+2) +log(x+4)]/log3 + log(x+2)/log(1/3) < 1/2log7/log (3^1/2)
[log(x+2) +log(x+4)]/log3 - log(x+2)/log(3) < 1/2log7/(1/2*log (3))
log(x+4)]/log3 < log7/log (3)
log(x+4)
Since, log is an increasing function, hence, x+4 <7
x<3
 
2. a = log 18 to the base 12
a = log18/log12
Also, b = log54/log24
 
Hence ab+5(a-b) = log18/log12 *  log54/log24+ 5( log18/log12 - log54/log24)
= log18/log12 *  log54/log24+ 5( log18/log12 - log54/log24)
=1
 

Answered by  | 16th Apr, 2013, 06:14: AM

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