Math Is Fun Forum

And now for Q1:

First some preliminaries:

If

and if

And now the question

All logs in base 7

using A becomes

Divide by x

Replace 1 by log(7) and simplify the logs

Using A again gives

I will replace log(x) with X

Now what does the graph of the left hand side function look like?

It has two components. The first is a standard power graph, going through (0,1) and rising thereafter.  The second is is similar but, as 3/7 is under 1, it will come down from large y values to the left of the y axis, again go through (0,1) and then drop off to zero as X gets larger.

Together the second component will dominate when X is negative; then the first will dominate.  So the graph will start high, drop down to (0,2) and at some stage rise again.  Note that (0,2) gives one solution to the question.

Differentiating using B

The ln(3/7) is negative so this will give one turning point. It must be a minimum because of the overall shape already established; (0,2) could be the minimum; otherwise there will be exactly one more solution.

I can see that X = 1 gives x = 7 leading to 11/7 + 3/7 = 14/7 = 2. So here is the second solution.

So the equation has two solutions, x = 1 and x = 7.

Bob

hi Mathegocart

Admin and Mods check the forum most days and deal with spammers and their posts. 

At the bottom of every post you should be seeing a 'Report' button that flags up any post of concern.  Admin and Mods get the report message when they log in and can jump to the post and do something about it.  Are you seeing that button?  Some forum features were lost when we moved to the new server and things that I can see are not necessarily visible to all. So please confirm if you have the report option and try it out by reporting this post for me. You should get a message space to give your reason, so please make it clear it's a test for me.

If you stumble on an edited post that isn't showing as a new post you should still be able to report it.

I've had a look through the options available to me on the forum.  I can switch off the information on the post side panel so that a members website doesn't show, but that would have apply to everyone, not just one member.  Similarly signatures can be turned off.  I'm very reluctant to do either of these as good members suffer along with the bad.

Bob

Thanks zetafunc,

These others had slipped past me so thanks for the info.  It does look like that word attracts them so I've cleaned up all, and removed a welcome from me that no longer applies. Would you mind editing out the 'e' word please.

Ganesh: I have left your posts but would prefer you take out the ones that now have no context. Thank you

Bob

hi Scuba_Joe

Welcome to the forum.

Compuhigh has asked us not to publish their copyright worksheets so I have removed the questions you posted.  They have better lawyers than we do so it's best not to upset them.

We will certainly help you with any topic in general but we won't do your homework for you.

For questions like these it is essential to include a diagram too.

Looking forward to helping once these matters are sorted out.

Bob

hi Mathegocart and Ganesh,

I noticed that keiraec had a link to an essay writing website.  I have deleted the link. Normally I would remove the poster too, but what do you think?

Bob

You need to devise a way to tell if COL + ROW = an EVEN number or an ODD number.  EVEN means a BLACK square. ODD means a WHITE square.

This calculation will achieve that:

total = col + row

value = total/2 - INT(total/2) where the division is a normal division with possibly a remainder, and INT yields the integer value of the answer ie. discards the fraction.

Look what happens when total is even and when it is odd.

total even: value = <another integer> - <same integer> = zero.  This happens because there is no fractional part to discard.

total odd: value = <an integer + 0.5> - <same integer without the 0.5> = 0.5.

So the value calculation enables you to detect ODD and EVEN as needed.

Bob

hi dummy39

Welcome top the forum.

I think this site https://www.mathsisfun.com/ is as good as any book and its FREE!

Bob

hi tony123

Thanks for another interesting puzzle. I think this is a proof but it's a bit messy.

Bob

Up to this point all is ok.  You can test this by evaluating the arithmetic.

This is ok because both sides evaluate to 1/4.

It's what happens next that is not correct.

If you just square root both sides you get

and that is not correct.  It's the careless square rooting that causes the false result.

Because a square root can be + or - the equation is valid if one of the signs is changed.

Take it as a warning not to square root without thinking about what is happening.

Calculators will only give a positive square root because they are programmed to give a single answer.  Some people regard the root sign as meaning this which is fine.  But it's always worth thinking about the negative case, because that might give the answer you want.

Bob

hi tony123

Thanks for another great exercise.

I'm still stuck with

Easy to show it's a possible value for x+y and the graph suggests it's unique, but I still cannot prove it properly.  Hint please.

Bob

hi CurlyBracket

That's the value of k that I got, well done!

I'd not seen your Latin phrase before so I googled and couldn't find 'regulum' but rather 'regulam'. 

It seems to be an abbreviated version of:

"exceptio probat regulam in casibus non exceptis" meaning the exception that proves the rule.

In maths that is bizarre because any exception (counterexample) means the rules hasn't been proved, so I investigated further.

Here's an example of what I think it really means:

A museum sign says "Entry free on Sundays".  It doesn't say what happens the rest of the week but most people would infer that there is a charge on the other 6 days.  So Sunday's rule is the exception that implies a rule that applies for the rest of the week.

A similar example would be "No parking here on market days".

Bob

Unless I've completely misunderstood, it has to be unique.  The number is 36 so the product begins 3 x 6 x <something> and has to result in 36.  There's only one solution to this equation:

3 x 6 times [letter x] = 36  => 18x = 36  => x = 36/18 = 2

Bob