# Solving Logarithmic Equations: Analytical Method

• joo
In summary, the analytical method for solving log equations involves using the product log function or approximation algorithms such as the method of Newton-Raphson. The Lambert W function is another option, but it is not commonly used. The graphical approach is still taught in university and is preferred for tests without access to calculators.
joo
What is the analytical method for solving log. eqs., like x=2^x/14 ?

In high school they only teach us the graphical approach =/

joo

What I meant was x=(2^x)/14, but I guess that doesn't really change the principle.

Thanks, I'll take a look at it, although I find myself doubtful.

There is no analytical solution to your problem. One can express the solutions using the product log function, but that is just another way of writing it, not a true analytic solution.

Welcome to PF, joo! In university they still use the graphical approach. ;)

In addition they use approximation algorithms, like the method of Newton-Raphson (which is based on a Taylor series expansion).

It's only the really bold ones in math that use the Lambert W function, which is a function that has only been invented to be able to write the solution to your equation.
As far as I know, no one really uses it.

The first solution for your equation is ##x=-{W(-\frac 1 {14} \ln(2)) \over \ln(2)} \approx 0.07525##.

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I like Serena said:
Welcome to PF, joo!

In university they still use the graphical approach. ;)

In addition they use approximation algorithms, like the method of Newton-Raphson (which is based on a Taylor series expansion).

It's only the really bold ones in math that use the Lambert W function, which is a function that has only been invented to be able to write the solution to your equation.
As far as I know, no one really uses it.

(HUMOR)

But Lambert used it and they made a movie on his life among sheep:

(/HUMOR)

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jedishrfu said:
(HUMOR)

But Lambert used it and they made a movie on his life among sheep:

(/HUMOR)

Oh! So the W comes from Walt Disney! :D

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Thank you for your replies ! I'll stick to the graphical solving for now then, since I will have no access to any calculators during my tests.

## What is the analytical method for solving logarithmic equations?

The analytical method for solving logarithmic equations involves using algebraic techniques to rewrite the logarithmic equation into an exponential form and then solving for the variable.

## When should the analytical method be used for solving logarithmic equations?

The analytical method should be used when the logarithmic equation has only one logarithm and the base of the logarithm is the same for both sides of the equation.

## What are the steps involved in using the analytical method to solve logarithmic equations?

The steps involved in using the analytical method to solve logarithmic equations are: 1. Rewrite the logarithmic equation in exponential form.2. Isolate the variable by applying properties of logarithms.3. Solve for the variable using algebraic techniques.

## Can the analytical method be used for all types of logarithmic equations?

No, the analytical method can only be used for logarithmic equations with one logarithm and the same base on both sides of the equation. For equations with multiple logarithms or different bases, other methods such as graphing or using a calculator may be more appropriate.

## Are there any tips for using the analytical method to solve logarithmic equations?

Yes, some tips for using the analytical method to solve logarithmic equations are: - Familiarize yourself with the properties of logarithms.- Double-check your answer by plugging it back into the original equation.- Be careful with negative solutions and make sure they make sense in the context of the problem.

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