Solve for x.

Question

4^x - 10•4^-x = 3

Answer 1

Subtract 3 from both sides:

4^x -10(4^-x) -3 = 0

 

I multiplied through by 4^x.

 

4^x(4^x -10*4^-x -3) = 4^x*0

 

4^2x - 10* 4^x-x - 3*4^x = 0

 

Since 4^x-x = 4⁰ = 1, rearrange:

 

4^2x -3(4^x) - 10 = 0

 

4^2x is the same as 4^(x)(2)

 

(4^x)² -3(4^x) - 10 = 0.

 

 

This is like a quadratic.  Set u = 4^x  and substitute.

 

u2 - 3u - 10 = 0

 

Factor:

 

(u-5)(u+2) = 0

 

 

Put the 4^x back in for u:

 

4^x - 5 = 0              4^x + 2 = 0

4^x = 5                   4^x = -2

 

ln 4^x = ln 5             No solution since there exists no power to which a positive 4 can be raised to get -2.

 

x ln 4 = ln 5

 

x = ln 5/ln 4

 

Also:

ln 4 =  ln 2²  = 2 ln 2

Answer 2

x = ln(5)/2*ln(2)