# log help

• Nov 17th 2005, 04:08 PM
floridaboatkid
log help
Been trying to figure this out but haven't had any luck

There is eactly one real number k for which the equation
(log3x)(log5x) =k
has only one real solution for x. What is the value of x.
• Nov 18th 2005, 01:49 AM
CaptainBlack
Quote:

Originally Posted by floridaboatkid
Been trying to figure this out but haven't had any luck

There is eactly one real number k for which the equation
(log3x)(log5x) =k
has only one real solution for x. What is the value of x.

I am guessing but I would bet ready money that your question should

"There is eactly one real number k for which the equation
(log_3(x))(log_5(x)) =k
has only one real solution for x. What is the value of x"

Where log_b denotes the log to base b.

So lets assume your question is:

"There is eactly one real number k for which the equation
$\displaystyle log_3 (x).log_5 (x)\ =\ k$
has only one real solution for x. What is the value of x"

as $\displaystyle log_b(x)\ =\ log_e(x)/log_e(b)$ your equation becomes:

$\displaystyle \frac{log_e (x).log_e (x)}{log_e(3).log_e(5)}\ =\ k$

or:

$\displaystyle (log_e(x))^2\ =\ k.log_e(3).log_e(5)$

Then the condition of the problem implies that $\displaystyle k\ =\ 0$
as otherwise there will be two solution for $\displaystyle x$, so:

$\displaystyle log_e(x)\ =\ 0$

or
$\displaystyle x\ =\ 1$

Quite Easily Done (aka QED :) )

RonL
• Nov 20th 2005, 02:30 PM
floridaboatkid
no they are base 10
the equation is written in log base 10
• Nov 20th 2005, 08:34 PM
CaptainBlack
Quote:

Originally Posted by floridaboatkid
the equation is written in log base 10

OK, then lets rewrite your problem so that it is unambiguous:

Quote:

There is exactly one real number $\displaystyle k$ for which the equation:
$\displaystyle (log_{10}(3x))(log_{10}(5x)) =k$
has only one real solution for $\displaystyle x$. What is the value of $\displaystyle x$.
Now this equation only makes sense with:
$\displaystyle x\ \epsilon\ (0,\ \infty).$
Also as:
$\displaystyle x\ \rightarrow\ 0\ ,\ (log_{10}(3x))(log_{10}(5x))\ \rightarrow\ \infty,$
and
$\displaystyle x\ \rightarrow\ \infty\ ,\ (log_{10}(3x))(log_{10}(5x))\ \rightarrow\ \infty.$

So an equation:
$\displaystyle (log_{10}(3x))(log_{10}(5x))\ =\ k$
has either multiple solutions, zero solutions, or when $\displaystyle k$ is the global
minimum of $\displaystyle (log_{10}(3x))(log_{10}(5x))$ exactly one solution.

So your problem becomes find the $\displaystyle x$ which corresponds to the global minimum of:

$\displaystyle y(x)\ =\ (log_{10}(3x))(log_{10}(5x))$

RonL
• Nov 21st 2005, 01:23 AM
CaptainBlack
Now our problem is to find the global minimum of:

$\displaystyle y(x)\ =\ (log_{10}(3x))(log_{10}(5x))$

The first thing we do is to sketch the graph of this
function. A couple of rather pathetic Excel plots of
this are shown in the attachment to this post.

From these plots it is clear that the minimum we seek
is the sort which is obtained by differentiating and
setting the resulting derivative to zero, that is we seek
the solution of:

$\displaystyle \frac{dy}{dx}\ =\ 0$

which should be a routine task, which I will leave to you.

RonL