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.

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- Nov 17th 2005, 04:08 PM #1

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- Nov 18th 2005, 01:49 AM #2

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Originally Posted by**floridaboatkid**

read somethink like:

"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 #3

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- Nov 20th 2005, 08:34 PM #4

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Originally Posted by**floridaboatkid**

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 $\displaystyle x$.

$\displaystyle x\ \epsilon\ (0,\ \infty).$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 #5

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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