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If x is a real number such that 2^{2x+3}=14, find \(2^x\).

 Apr 19, 2018
 #1
avatar+17338 
+2

2^{2x+3}=14     Take log of both sides

 

(2x+3) Log 2 = log 14     solve for 'x'

 

          

x = (log 14/log2 -3  )/2   =  .403677461        Then 2^x = 1.322876

 Apr 19, 2018
 #2
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0

please help me. im stuck.

Guest Apr 19, 2018
 #3
avatar+771 
+2

2(2x+3)=14

The first thing you do is take a log base 2 of both sides, as exponentials and logarithms are inverses. I got the base of 2 from what the 2x+3 is raised to, which is a 2.

\(log_2(2^{2x+3})=log_2(14)\)

Since the base of the logarithm and the base of the exponential are the same, they cancel, leaving only 2x+3.

\(2x+3=log_2(14)\)

Subtract 3 from both sides.

\(2x=log_2(14)-3\)

Divide by 2

\(x=\frac{log_2(14)-3}{2}\)

\(x=0.403677461\)

 

So, 2x is equivalent to \(2^{0.403677461}\).

\(2^{0.403677461}=1.322876\)

 

So, ElectricPavlov is right about the final answer, but I wanted to reiterate the steps in a little more detail, just for clarity's sake.

 Apr 19, 2018
 #4
avatar+98198 
0

Thanks, EP  and AT   !!!!!!

 

 

 

cool cool cool

CPhill  Apr 19, 2018
 #5
avatar+27578 
+1

Since the question asks for 2x, not x, there is no need to take logs:

 

2(2x+3) = 14

22x.23 = 14

22x = 14/8   (since 23 = 8)

(2x)2 = 14/8

2x = (14/8)1/2

2x ≈ 1.322876

 Apr 19, 2018

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