SOLUTION: Solve this equation. 5^x^3 = 25^x x =___________ ____(smallest value) x = x =___________ _____

SOLUTION: Solve this equation. 5^x^3 = 25^x x =_____ (smallest value) x = x =_ __(largest value

Question 116625: Solve this equation.
5^x^3 = 25^x
x =________________(smallest value)
x =
x =__________________(largest value
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
In case you can still use an answer to this problem, here's how you can do it.
.
The biggest lesson that you can learn from this problem is that anytime you see a variable used
as an exponent, you should think of the possibility of using logarithms to solve the problem
because if you take the logarithm of such a quantity, the exponent can be brought out as the
multiplier of the logarithm. This is true regardless of base in the logarithm. For example:
.
is equal to
.
In this problem you are given the equation:
.

.
If you take the Logarithm of both sides (any base will do, but just for simplicity let's
use base 10), the equation becomes:
.

.
bringing out the exponents and making them multipliers converts the problem to:
.

.
But notice on the right side that 25 can be replaced by making the equation become:
.

.
Applying the exponent property again on the right side, the 2 comes out as a multiplier and
you have:
.

.
Notice now that is a factor of both sides, and it is just a number that you
can get using a calculator. [Should you be interested, that number is 0.698970004].
If both sides of the equation are divided by the logarithmic term just
cancels on both sides and the equation is reduced to:
.

.
Get rid of the 2x on the right side by subtracting 2x from both sides to get:
.

.
Factor the left side and you get:
.

.
This equation will be true if either of the factors on the left side equals zero because then
the left side would involve a multiplication by zero and that zero multiplier would make
the entire left side equal to zero ... therefore equaling the right side.
.
So setting the first factor equal to zero just gives:
.

.
Then setting the second factor equal to zero gives:
.

.
Add 2 to both sides and you get:
.

.
Solve for x by taking the square root of both sides and you have:
.
and
.
In summary, the three answers to this problem are:
.

.
If you substitute each of these three values back into the original equation, you will find
that the equation is true.
.
For example: Start with the original equation:
.

.
Substitute 0 for x and you have:
.

.
but you know that that when any quantity is raised to the zero power the answer is 1. So this
equation reduces to:
.

.
And this is obviously true ... so x = 0 is a good solution.
.
Next start with the original equation and substitute for x.
.
The original equation is:
.

.
Substitute for x and get:
.

.
On the right side replace 25 with and the equation becomes:
.

.
On the left side cubing is equivalent to and
so cubing results in . This makes the equation become:
.

.
Finally on the right side, the power rule of exponents tells you to multiply 2 times
and this makes the equation become:
.

.
Note that the left side now equals the right side, so is a good solution
to the problem.
.
You can do the proof for . It works very similar to the one for .
.
I know this is late, but the lessons involved in solving this problem are hopefully in
time to help you understand the concepts involved in getting a solution.
.
Happy holidays to you and yours ...
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SOLUTION: Solve this equation. 5^x^3 = 25^x x =___________ ____(smallest value) x = x =___________ ______(largest value

SOLUTION: Solve this equation. 5^x^3 = 25^x x =_____ (smallest value) x = x =_ __(largest value