Log and > problem

  • #1

Homework Statement



it wants the sum of all solutions

2 different problems 1. and 2.

1. 2^x + 2^(x+1) + 2^(x+2) = 2^6.

2. 8/(3^x +2)>=3^x

>= meaning equal or greater than 3^x

Homework Equations



should i come up with a summa equation?

The Attempt at a Solution



i switched 3^x=y and solved it but i only got 1 solution and apparently there are more. if anyone can help. thanks.
 

Answers and Replies

  • #2
eumyang
Homework Helper
1,347
10
1. 2^x + 2^(x+1) + 2^(x+2) = 2^6.

Note that
[tex]2^{x+2} = 2^2 \cdot 2^x = 4 \cdot 2^x[/tex]
and
[tex]2^{x+1} = 2^1 \cdot 2^x = 2 \cdot 2^x[/tex].

So
[tex]2^x + 2^{x+1} + 2^{x+2} = 2^x + 2 \cdot 2^x + 4 \cdot 2^x = 7 \cdot 2^x[/tex].
Take it from there.
 
  • #3
34,965
6,717

Homework Statement



it wants the sum of all solutions

2 different problems 1. and 2.

1. 2^x + 2^(x+1) + 2^(x+2) = 2^6.

2. 8/(3^x +2)>=3^x

>= meaning equal or greater than 3^x

Homework Equations



should i come up with a summa equation?

The Attempt at a Solution



i switched 3^x=y and solved it but i only got 1 solution and apparently there are more. if anyone can help. thanks.
In #1, the equation is the same as 2x + 2*2x + 4*2x = 26. Can you solve that one?

In #2, 3x > 0 for all x, so 3x + 2 > 2 for all x.
Multiplying both sides of the inequality by 3x + 2 won't change the direction of the inequality. If you do this, you get an inequality that is quadratic in form. Can you show us what you did?
 

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