SOLUTION: The demand function for a certain commodity is given by p=80e exponent fraction -q/2 Write q as a function of p

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Question 70663: The demand function for a certain commodity is given by p=80e exponent fraction -q/2 Write q as a function of p
Found 2 solutions by funmath, bucky:
Answer by funmath(2933) About Me  (Show Source):
You can put this solution on YOUR website!
The demand function for a certain commodity is given by p=80e exponent fraction -q/2 Write q as a function of p
p=80e%5E%28-q%2F2%29
p%2F80=80e%5E%28-q%2F2%29%2F80
p%2F80=e%5E%28-q%2F2%29
ln%28p%2F80%29=ln%28e%5E%28-q%2F2%29%29
ln%28p%2F80%29=-%28q%2F2%29ln%28e%29 ln(e)=1, so
ln%28p%2F80%29=-q%2F2
-2ln%28p%2F80%29=-2%28-q%2F2%29
-2ln%28p%2F80%29=q
highlight%28q=-2ln%28p%2F80%29%29%29
Happy Calculating!!!!

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
p+=+80e%5E%28-q%2F2%29 Solve for q.
.
This is what I understand your problem to be. Assuming this is correct, you can take the ln
(natural logarithm which has the base e) of both sides and the problem becomes:
.
+ln%28p%29=+ln%2880%2Ae%5E%28-q%2F2%29%29
.
But the logarithm of a product is equals the sum of the logarithms of the two terms being
multiplied. Therefore, we can split the right side into the sum of two logarithms as follows:
.
+ln%28p%29=+ln%2880%29%2B+ln%28e%5E%28-q%2F2%29%29
.
Subtract ln(80) from both sides to get:
.
+ln%28p%29+-+ln%2880%29+=++%2B+ln%28e%5E%28-q%2F2%29%29
.
By the rules of logarithms, the difference of the logarithms of two quantities can be
re-written as the logarithm of the quotients of the quantities. This translates to:
.
+ln%28p%29+-+ln%2880%29+=+ln%28p%2F80%29
.
Substituting this as a replacement for the left side results in:
.
+ln%28p%2F80%29+=+ln%28e%5E%28-q%2F2%29%29
.
Then by a rule of exponents in logarithms, the exponent of a term in a logarithm becomes the
multiplier of the logarithm of the term on the right side. In this case %28-q%2F2%29
becomes the multiplier of ln(e) and the right side of the equation is changed as shown below:
.
+ln%28p%2F80%29+=+%28-q%2F2%29%2Aln%28e%29
.
But ln(e) = 1, and when this substitution is made the equation becomes:
.
+ln%28p%2F80%29+=+%28-q%2F2%29
.
Multiply both sides of the equation by -2 and the equation becomes:
.
+-2%2Aln%28p%2F80%29+=+q
.
This is the answer, but there is one additional constraint. The value of p must be greater
than zero or else you would be taking the ln of a negative number or zero and those are
outside of the allowed values of numbers that the ln function can operate on.
.
Hopes this gives you some additional insight about the subject of logarithms and natural
logarithms in particular.