Can you define the function?

Algebra Level 4

y = log ⁡ 7 − a ( 2 x 2 + 2 x + a + 3 ) \large y=\log _{ 7-a }{ (2{ x }^{ 2 }+2x+a+3) }

If y y above is defined for all real x x , then find the number of possible integral values of a a .

7 9 10 8

Yash Choudhary by Yash Choudhary , 22, India

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2 solutions

Yash Choudhary
Feb 28, 2015

A log function is define when the number inside it is positive and base is positive as well as not equal to one. = = > 2 x 2 + 2 x + a + 3 > 0 D i s c r i m i n e n t s h o u l d b e n e g a t i v e = = > 2 2 − 4 × 2 ( a + 3 ) < 0 = = > 4 − 8 a − 24 < 0 = = > 2 a + 5 > 0 = = > a > − 5 2 A l s o b a s e s h o u l d b e g r e a t e r t h a n o n e = = > 7 − a > 1 = = > a < 6 H e n c e , i n t e g r a l v a l u e s o f a a r e − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , 5. T h e r e f o r e , a n s w e r i s 8 . ==>\quad 2{ x }^{ 2 }+2x+a+3>0\\ Discriminent\quad should\quad be\quad negative\\ ==>\quad { 2 }^{ 2 }-4\times 2(a+3)<0\\ ==>\quad 4-8a-24<0\\ ==>\quad 2a+5>0\\ ==>\quad a>-\frac { 5 }{ 2 } \\ Also\quad base\quad should\quad be\quad greater\quad than\quad one\\ ==>\quad 7-a>1\\ ==>\quad a<6\\ Hence,\quad integral\quad values\quad of\quad a\quad are\quad -2,-1,0,1,2,3,4,5.\\ Therefore,\quad answer\quad is\quad \boxed { 8 } .

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I think that you should use \Rightarrow to get ⇒ \Rightarrow

⌣ ¨ \ddot\smile

A Former Brilliant Member - 6 years, 3 months ago

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Ty bro i'll use that next time.

Yash Choudhary - 6 years, 3 months ago

I think that if you are writing that the base should be greater than 1 than the values of 7 − a 7-a which are lying between 0 and 1 are excluded but as here in the question only integral values are desired so it doesn't matters but logically it should be that the value of 7 − a 7-a should be greater than 0 but not equal to 1.

Aditya Tiwari - 6 years, 3 months ago

Would 0 also be a solution for a? Why would it not have 9 solutions, since 0 works?

Michael Boyd - 4 years, 3 months ago
Hiroto Kun
Mar 3, 2017

I think that if you are writing that the base should be greater than 1 than the values of which are lying between 0 and 1 are excluded but as here in the question only integral values are desired so it doesn't matters but logically it should be that the value of should be greater than 0 but not equal to 1.

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