Logarithm question: Solve log_5(x - 4) = log_7(x)

[gymnastqueen]

The question I have is log[sub:e4kdqu1h]5[/sub:e4kdqu1h](X-4)=log[sub:e4kdqu1h]7[/sub:e4kdqu1h]X and it is asked that you solve for X, I've tried it a bunch of ways but just can't seem to finish it correctly. If anyone can help it would be much appreciated!

 

[r_yackel]

Re: Logarithms question

show me some of the work you have done.

 

[Deleted member 4993]

Re: Logarithms question

The question I have is log[sub:7y9isrus]5[/sub:7y9isrus](X-4)=log[sub:7y9isrus]7[/sub:7y9isrus]X and it is asked that you solve for X, I've tried it a bunch of ways but just can't seem to finish it correctly. If anyone can help it would be much appreciated!

First perform a chage-of-base operation to make the bases same.

 

[gymnastqueen]

I have tried the change of base formula, but I can never get to the answer I get when I graph it on my calculator to verify (11.548219). I'm not sure if when I use this formula my logs should be to a base of 35 or of 10. I have tried both, but neither gives me the correct answer and as I continue through the work my calculations just get more and more bungled and complicated rather than being simplified so I can solve. I guess my real problem is getting both X's on the same side, without them cancelling eachother. You can see the work I've done below. Please help if you can!

log[sub:u0xqrqlc]5[/sub:u0xqrqlc](X-4) = log[sub:u0xqrqlc]7[/sub:u0xqrqlc]X
log(X-4)/log5 = logX/log7
log(X-4)-log5 = logX-log7 --->here I'm uncertain, should I split log(X-4) into logX-log4?
log(x-4)-logX = log5-log7
log(X-4)/logX = log5/log7 --->this is as far as I get and then my X's cancel and I end up with log4 = log5/log7 which clearly isn't right

 

[stapel]

I have tried the change of base formula, but I can never get to the answer I get when I graph it on my calculator to verify (11.548219).

I will guess that you graphed Y1 = (ln(X-4))/(ln(5)) and Y2 = (ln(X))/(ln(7)) and found the intersection point. I get the same value.

I'm not sure if when I use this formula my logs should be to a base of 35 or of 10.

The point of the change-of-base formula is that the base can be anything you like! :wink:

should I split log(X-4) into logX-log4?

Since this is not a property of logarithms (the rule is that log[sub:3o55l39f]b[/sub:3o55l39f](m) - log[sub:3o55l39f]b[/sub:3o55l39f](n) = log[sub:3o55l39f]b[/sub:3o55l39f](m/n); that is, the subtraction outside the logs becomes a division inside a log), you cannot legitimately take this step. :shock:

What solution method are you "supposed" to use for this?

Eliz.

 

[gymnastqueen]

It doesn't give a solution method, it just says solve for x and round to 4 decimal places... I'm not sure what to do lol. You've already been more help than my teacher he just says we have to figure it out on our own, which obviously isn't working for me :roll: he won't even answer my questions so I really appreciate your help you've made things a lot clearer already. So if log(x-4) is divided by x it just leaves 4 right? Maybe it's just my math that is wrong? Would I be able to make log(x-4) into logx/4 then?

 

[stapel]

if log(x-4) is divided by x it just leaves 4 right? Maybe it's just my math that is wrong? Would I be able to make log(x-4) into logx/4 then?

You might want to review the log rules again. There is no rule that does what you state (above).

Eliz.

 

[gymnastqueen]

I know the rules I've been over them 20 times... I just can't find any of them that help me solve this equation that's why I'm grasping at straws trying to review everything I could have maybe done wrong. Nobody can solve this equation, so I'm just trying to find anything that may have been missed, I'm looking for the secret key if you will.

 

[pka]

l

I have tried the change of base formula, but I can never get to the answer I get when I graph it on my calculator to verify (11.548219).(

You have found the correct root.
But the following contain several mistakes.
log(X-4)/log5 = logX/log7 & log(X-4)-log5 = logX-log7 are both mistaken.
\(\displaystyle \log \left( {\frac{{x - 4}}{5}} \right) ={ \log (x - 4) - \log (5)\,\mbox{but} \,\frac{{\log (x - 4)}}{{\log (5)}} \ne \log (x - 4) - \log (5)}\).

The most mathematically correct way to rewrite your problem is:
\(\displaystyle \ln (7)\ln (x - 4) - \ln (5)\ln (x) = 0\).

Depending on your mathematical level, there are several numerical methods we can use to solve that equation. But the are no simple solutions.