First note that because logarithms are defined only for numbers > 0 from the left hand side you know x >0, and from the right hand side you know that x <1. Now consider how you could make the left hand side = 1. If you use that value for x in the right hand side, what do you get?

According to what you said I got x=9/10, which is the correct answer, but I don't understand why to make the left hand side = 1. This is the only method that can be applied?

And what if the roots are > 1 ? Like in this case:

I can't find a closed form solution that you can solve. But one line of reasoning that helps lead to this solution goes like this:

1. Let a = 3/10. The formula you're tryng to solve is:



2. Apply the conversion:


3. Use the fact that if
then :



4. You know that the left hand side is less than 1/3, since x < 1. For the right hand side to be less than 1/3 means that:



5. So now you know that 0.895 < x < 1.

At this point you can see that x = 9/10 makes for a simple answer. The alternative is to use a numerical technique to estimate the value of x. Usoing something like Newton's method zeroes in on x = 9/10 very quickly.

Quote:

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Originally Posted by lemon14

And what if the roots are > 1 ? Like in this case?

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For just consider numbers that would work nicely. It'll take some trial and error. For example, in considering log base 5 the kinds of numbers that work "nicely" are 5, 5^2, 5^3, etc. Try them out and see what you get. For example, if the right hand side is set to , then 2x+7 = 5 and x = -1. Now see if x = -1 satisfies the equation. Clearly it doesn't. So try the value of x that gives the next "nice" result for the right hand side of and see what happens.