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How to solve 4.466^x -2(2.331)^x + 1 = 0 ?

 Jul 28, 2014

Best Answer 

 #3
avatar+128079 
+5

Here's a graph of this one.........

 

The solutions (roots) are: x = 0 (as Alan noted) and x ≈ .354336....Notice how the lead term "over-powers" the graph very quickly as x moves away from zero in a positive direction. Also, as x gets more and more negative, the first two terms → 0 and the graph has a limit of 1 on the left hand side.....

 

 Jul 29, 2014
 #1
avatar+4473 
0

4.466^x -2(2.331)^x + 1 = 0

4.466^x - 4.662^x = -1 -->

-0.196^x = -1 -->

log(-0.196^x) = log(-1) -->

Properties of logs allow: x log(-0.196) = log(-1) -->

x = log(-1) / log(-0.196) = Error

However, when plugging in values such as x = 0.02 into -0.196^x = -1, we obtain a number that is close to -1, -0.9679326095185039.

 Jul 28, 2014
 #2
avatar+33603 
+5

If only it were that simple; unfortunately ax - bx doesn't equal (a-b)x in general!

4.466x - 2*2.331x + 1 = 0 has one immediately obvious solution; namely x = 0, because:

4.4660 - 2*2.3310 + 1 = 1 - 2*1 + 1 = 0.

I can't see any way of getting another (real number) solution except by a numerical method.  A straightforward, though rather inefficient way is just to rearrange the equation as shown below, guess an initial value and then iterate until the iterates converge.  It takes about 100 iterations to get 5 decimal places!

iterations

 Jul 28, 2014
 #3
avatar+128079 
+5
Best Answer

Here's a graph of this one.........

 

The solutions (roots) are: x = 0 (as Alan noted) and x ≈ .354336....Notice how the lead term "over-powers" the graph very quickly as x moves away from zero in a positive direction. Also, as x gets more and more negative, the first two terms → 0 and the graph has a limit of 1 on the left hand side.....

 

CPhill Jul 29, 2014

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