SOLUTION: Dear tutor thank you (in advance for answering my question) The question I wanted to ask is How do you solve the equation {{{e^x=4sqrt(e^x)-3}}} I knew that i need to use y t

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Question 107579: Dear tutor thank you (in advance for answering my question)
The question I wanted to ask is
How do you solve the equation

I knew that i need to use y to substitute the equation but how?
Found 2 solutions by stanbon, bucky:
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
e^x=4sqrt(e^x)-3
e^x+3 = 4sqrt(e^x)
Square both sides to get:
e^2x + 6e^x + 9 = 16e^x
e^2x -10e^x + 9 = 0
(e^x-9)(e^x-1)=0
e^x=9 or e^x=1
x =ln9 or x=ln1
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Check these answers in e^x = 4sqrt(e^x)-3
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Checking x=ln(9)
e^(ln9)= 4sqrt(e^(ln9))-3
9 = 4*3-3
9 = 9
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Checking x = ln1
e^ln1 = 4sqrt(e^ln1)-3
e^0 = 4sqrt(1)-3
1 = 4-3
1 = 1
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Both solutions are good.
================
Cheers,
Stan H.
Cheers,
Stan H.

Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
Given:
.

.
Just to make things easier to see, for the time being let's substitute A for .
When you make that substitution, the equation becomes:
.

.
Let's now get rid of the -3 on the right side by adding +3 to both sides. When you do that,
the +3 cancels the -3 on the right side ... and the left side gets a +3. The equation
then becomes:
.

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By doing the above work we have isolated the square root term on the right side. This makes
it easier to do the next step which is to square both sides. On the left side squaring (A + 3)
results in and on the right side, squaring results in
. So, at this point our equation has become:
.

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Now get rid of the 16A on the right side by subtracting 16A from both sides. Subtracting
16A from the right side results in zero. And on the left side, the minus 16A combines with
the +6A to give -10A. So the equation is reduced to:
.

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The left side of this fraction factors into and so the equation becomes:
.

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Notice that this equation will be true if either of the two factors on the left side equals zero
(because multiplication by a zero on the left side makes the left side become zero and
therefore it is equal to the right side.) So we can find two answers for A by setting the
factors (one at a time) equal to zero. Setting the first factor ... (A - 9) ... equal to zero
results in:
.

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and when you add 9 to both sides you get
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Next set the second factor ... (A - 1) ... equal to zero:
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And when you add 1 to both sides you get
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So we have two answers ... A = 9 and A = 1. But don't forget that we defined A as equal to .
So let's replace A in each of these answers with . When we do that we get:
.
and
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Let's solve for x using the first of these answers ....
.
Start with
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Take the natural log of both sides to get:
.

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By the rules of logarithms the exponent x on the left side can be brought out as a multiplier
and the equation then becomes:
.

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But ln(e) equals 1, so on the left side you can replace ln(e) with 1 and then the left side
becomes x*1 or just x. The right side is ln(9) and you can use a scientific calculator to
determine that ln(9) is 2.197224577. Substituting these two values results in the equation
being reduced to:
.

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Now let's work on the second factor which is:
.

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This is a little easier because that any number (e is just a number) raised to the zero power
is equal to 1. Therefore because it makes become and that
is equal to 1 which is equal to the right side.
.
So this problem has two answers ... x = 2.197224577 and x = 0
.
Hope this clarifies things for you a little and that you are now able to see how the problem
might be worked.
.