help please

 

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

 

In the problem we have 2^{(2x+ 3)} = 14   

 

So:  To what power do we raise 2 in order to produce 14.  

 

The logarithm of a number is the power to which a particular  

base must be raised to produce a specific number.  

 

Example:    2³  =  8         so  log₂(8) = 3   

                10²  =  100     so  log₁₀(100) = 2  

 

The calculator in my iPhone has a button that will calculate logs.   

 

So, log₂(14) = 3.8073549 and the decimal fraction keeps going  

but that much of it will get close enough for our purpose.   

 

So 2^(3.8073549)  =  14 

 

This exponent is the value needed for the exponent in the problem.  

 

2^(2x+3) = 14    therefore  2x + 3  =  3.8073549 

                                       2x        =  0.8073549    

                                         x        =  0.4036775   

 

                                         2^x  =  2^(0.4036775)   

                                         2^x  =  1.32288   

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