Solving this logarithmic equation analytically

[Mr Davis 97]

I have the equation $3x + \log_5x = 378$.

Is there an analytical way to solve for x? Or for this equation are we forced to just try possible values, such as powers of 5?

 

[S.G. Janssens]

It is a transcendental equation, but in this case you can indeed find its solution by trying some small powers of $5$.

 

[mathman]

To get a good first guess use 3x=378 or x=126. You will quickly get to the solution (x=125).

 

[micromass]

Note that in general such equations do not have very nice solutions. But it appears you got lucky here. Normally, you would need to resort to numerical answers.

 

[Stephanie Riggs]

You can try using Lambert W function or https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_3_11_11.pdf to solve this equation. Unfortunately, I don't quite fully understand these methods (and I hope my test won't have these..I still don't quite understand these methods even after trying to solve online algebra practice tests..the methods proposed above are easy but I don't think they'll come in handy on a test) but I've seen such problems being solved with the help of Lambert W function. So, if you are better at it you can try doing it. And here is an equation similar to yours.

 

[chiro]

Hey Mr Davis 97.

Aside from a numerical solution, you will have to find a transformation so that the transformation allows one to get a solution in some basis (like integers, rationals, or some function of other quantities).

For this problem you will have to find a transformation u(x) to take x to u (and that preserves the inequality) where the transformation has a classification so that the solution is in closed form.

You could start by noting that log_a(x) + log_a(y) = log_a(xy) and mucking around with that from here on in.

 

[BaileyBelmont]

Hello!
I know the only one analytical way to solve it - with plots. Here is attached the plot of two parts of equation( you can do the same one with your favorite graphics builder( i prefer wolfram ar https://handmadewritings.com). On this plot you can easily see the only point that belongs to these two functions at the same time and it is (125, 378). That's how you can get your answer 125!

 

[Mark44]

I know the only one analytical way to solve it - with plots.

That's a graphical solution, not an analytic solution, which means finding a solution by algebraic means.