Solving this logarithmic equation analytically

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  • Thread starter Mr Davis 97
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In summary, the equation ##3x + \log_5x = 378## can be solved analytically by trying small powers of 5 as initial guesses, such as 3x = 378 or x = 126. This leads to the solution x = 125. However, this is a rare case of a transcendental equation having a nice solution. In general, numerical methods or transformations may be needed to solve such equations. One possible transformation is using logarithmic properties, such as log_a(x) + log_a(y) = log_a(xy), to simplify the equation. Another approach is using graphical methods to find the intersection point of the two functions. However, this is not considered an analytical solution.
  • #1
Mr Davis 97
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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?
 
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  • #2
It is a transcendental equation, but in this case you can indeed find its solution by trying some small powers of ##5##.
 
  • #3
To get a good first guess use 3x=378 or x=126. You will quickly get to the solution (x=125).
 
  • #4
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.
 
  • #5
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.
 
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  • #6
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.
 
  • #7
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!
2f08e41e8ab144e2a190d320347382ed.png
 
  • #8
BaileyBelmont said:
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.
 

1. What is a logarithmic equation?

A logarithmic equation is an equation that contains one or more logarithmic terms. Logarithmic functions are the inverse of exponential functions and are used to solve for unknown variables in exponential equations.

2. How do you solve a logarithmic equation analytically?

To solve a logarithmic equation analytically, you need to isolate the logarithmic term on one side of the equation and the other terms on the other side. Then, you can use the properties of logarithms, such as the product rule, quotient rule, and power rule, to simplify the equation and solve for the variable.

3. What are the common mistakes to avoid when solving a logarithmic equation?

Common mistakes to avoid when solving a logarithmic equation include forgetting to apply the properties of logarithms, making algebraic errors, and not checking for extraneous solutions. It is important to carefully work through each step and double-check your answer to avoid these mistakes.

4. Can I use a calculator to solve a logarithmic equation?

Yes, you can use a calculator to solve a logarithmic equation. However, it is important to remember to use the correct logarithmic function on your calculator, such as log or ln, and to check your answer manually to avoid any errors.

5. When is it necessary to use logarithmic equations in science?

Logarithmic equations are commonly used in science to model various phenomena, such as population growth, radioactive decay, and pH levels. They are also used in data analysis and in solving exponential growth or decay problems, making them an important tool in many scientific fields.

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