Logarithm problem once again...

[nigahiga]

I cannot change the bases into a common base for the life of me. Can someone give a suggestion as to how I can proceed in this problem?

 

[pka]

I cannot change the bases into a common base for the life of me. Can someone give a suggestion as to how I can proceed in this problem?

Yes, just look at the question, the solution is obvious.
HINT: \(\displaystyle 2+5=7\).

 

[Deleted member 4993]

One solution I see by observation is:

4^log(x) = 25^log(x) = x

we know log(1) = 0 and

4⁰ = 25⁰ = 1 → x =1

 

[HallsofIvy]

Which works precisely because 2+ 5= 7 which is just what pka said!

 

[nigahiga]

I don't really understand I know the answer is 1 but I cannot find what substitution or method to use

 

[pka]

I don't really understand I know the answer is 1 but I cannot find what substitution or method to use

The whole point is: There is no method other than simple inspection.

You should know that if \(\displaystyle b>0\) then \(\displaystyle \log_b(1)=0\).

 

[JeffM]

I don't really understand I know the answer is 1 but I cannot find what substitution or method to use

First of all you did not give us the actual question, which I suspect is a trick question phrased something like

Find a solution to the following equation.

That is you are not asked to find all solutions (whether there are more than one or not I do not know).

Furthermore when you see 2a + 5b = 7x, you know by inspection that a solution to any such equation is at (1, 1, 1).

Of course, a and b are functions of x so that observation is helpful only if a = f(1) = 1 = g(1) = b. So you test, and lo and behold they do.

 

[nigahiga]

First of all you did not give us the actual question, which I suspect is a trick question phrased something like

Find a solution to the following equation.

That is you are not asked to find all solutions (whether there are more than one or not I do not know).

Furthermore when you see 2a + 5b = 7x, you know by inspection that a solution to any such equation is at (1, 1, 1).

Of course, a and b are functions of x so that observation is helpful only if a = f(1) = 1 = g(1) = b. So you test, and lo and behold they do.

I apologize for the ambiguity. Thank you!

 

[lookagain]

I don't really understand I know > > > the < < < answer is 1, \(\displaystyle \ \ \ \) No, one of the answers is 1. There is another answer.

but I cannot find what substitution or method to use.

.* \(\displaystyle Look \ \ at \ \ x \ \ = \ \ 0.1 \ \ = \ \ 1/10 \ \ \ as \ \ another \ \ answer. \ \)*