Solving Logarithmic Equations: 2 Problems

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Homework Help Overview

The discussion revolves around two logarithmic equations that the original poster is attempting to solve. The first problem involves the expression log 7 (x^3 + 27) + log 7 (x + 3), while the second problem concerns the equation log5 (5^(1/x) + 125) = log 5 (6 + 1 + 1/(2x)).

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to manipulate the first equation by combining the logarithmic terms and setting it equal to a power of 7. They express uncertainty about how to proceed from there. For the second problem, they similarly attempt to rearrange the logarithmic expression but also express difficulty in progressing further.
  • Some participants question the clarity of the original problems, suggesting there may be typos or missing elements, particularly in the second problem where the equation may not be properly formed.
  • Others suggest factoring techniques and simplifications, such as recognizing the sum of cubes in the first problem and turning logarithmic expressions into exponential forms in the second.

Discussion Status

The discussion is ongoing, with various participants offering insights and suggestions for both problems. There is no explicit consensus on the correct interpretation of the problems, but several productive lines of reasoning have been explored, particularly regarding the first problem's structure and potential simplifications.

Contextual Notes

Participants note the presence of potential typos in the original equations, which may affect the interpretation and approach to solving them. The original poster has expressed urgency for assistance, indicating a time constraint in resolving these problems.

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Homework Statement


I have 2 problems ,which are:
1) Solve :log 7 (x 3+27) + log 7 (x+3)
2) Solve for x:log5 ( 5 1/x+ 125) log 5 6+1+1/2x


Homework Equations





The Attempt at a Solution


The first one tried like this :
log 7 { (x 3+27)/(x+3) }= 2
or 7 2 = (x 3+27)/(x+3)
But I don't know how to proceed beyond it please help!
in second one
log5{( 5 1/x+ 125) /6}=1+1/2x
but I can't proceed here too.
I need an urgent help!
Thank You!
 
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Please check the original problems as I see typos abound. For instance, in the first problem,

log 7 (x 3+27) + log 7 (x+3)

...I see no right hand side. Also, if what you have above is correct, then you're next step...

log 7 { (x 3+27)/(x+3) }= 2

...is already wrong. (Is is supposed to be divide?)

Finally, if the last step above is supposed to have a divide, then review factoring special products -- specifically, the sum of two cubes.


69
 
For the first one, you could multiply through by x+3 to give

[tex]49(x+3)=x^3+27[/tex]

and then solve that cubic by simplifying and turning it into the form [tex]ax^3+bx^2+cx+d=0[/tex]
but cubics are hard to solve, an easier way would be to realize that [tex]x^3+27[/tex] is a sum of two cubes, mainly [tex]x^3+3^3=(x+3)(x^2-3x+9)[/tex] which means you can cancel the x+3 from the numerator and denominator, giving you only a quadratic to solve, which is much simpler. Remember to look back at the original question to figure out what the domain of x is.

For the second, do the same thing as you did above, turn the log into powers such that

[tex]\frac{5^{\frac{1}{x}}+125}{6}=5^{1+\frac{1}{2x}}[/tex]

Now simplify the right hand side. You should be able to factorize out 51/x and then solve.
 
1/2" said:

Homework Statement


I have 2 problems ,which are:
1) Solve :log 7 (x 3+27) + log 7 (x+3)
2) Solve for x:log5 ( 5 1/x+ 125) log 5 6+1+1/2x

I agree with eumyang. Is the question to simply the above expressions? Also, with the second problem, are you missing any brackets/parenthesis at all?

The Bob
 
At least in the second, 1/2" specifically says "solve for x". There are no equations to be solved!
 
HallsofIvy said:
At least in the second, 1/2" specifically says "solve for x". There are no equations to be solved!

1/2" said:
2) Solve for x:log5 ( 5 1/x+ 125) log 5 6+1+1/2x

...

log5{( 5 1/x+ 125) /6}=1+1/2x

It's an honest mistake to mix up + and = considering they're the same key on the keyboard, but he did have lots of typos so it could have been re-interpreted as anything.

And for the first equation, let's just assume he started to solve the correct problem, i.e. f(x)=2.
 
Thanks for the help but I figured it out!
Thanks everyone!
 

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