How to solve equation using Function Calculator

Many modern calculators can evaluate expressions, but a programmable calculator allows you to go much further. 

You can create your own mathematical functions, automate calculations, and even build numerical solvers.

In this post, I will demonstrate how to create a custom function that solves the equation

using a programmable Android calculator that supports scripting.

This example shows how powerful programmable calculators can be when solving equations that cannot be easily rearranged algebraically.

1. Understanding the Equation

For example  We want to solve:

This equation cannot be solved with simple algebra. 

Instead, we rewrite it using logarithms:

x*ln(x)=ln(k)

Solving this equation is equivalent to solving the above equation.

2. Using the Newton Method

To find the root, we use the Newton–Raphson method, a numerical technique that iteratively improves an initial guess.

The update formula is: 

(next x)=x-f(x)/df(x)

For our function:

The derivative df(x) is  ln(x)+1

So the iteration becomes 

(next x)=x-(x*ln(x)-ln(k))/(ln(x)+1)

3. Implementing the Function in the Calculator

Below is the script used in the programmable calculator.

def xPOWx(k,guess){

    var x=guess;

    var tolerance=0.0000000001;

    var iterations=1000;

    var found=false;

    for(var i=0;i<iterations;i=i+1){

        var f=x*ln(x)-ln(k);

        var df=ln(x)+1;

        if(abs(df)<10^-12){

            break;

        }

        var nextX=x-f/df;

        if(abs(nextX-x)<tolerance){

            x=nextX;

            found=true;

            break;

        }

        x=nextX;

    }

    var noAnswer=-515151.515151;

    if(guess==x||found==true){

        return x;

    }

    else{

        return noAnswer;

    }

}

Copy above code and paste it in the Factory page of the calculator like below image and click triangle shaped run button and click save button.

4. How the Script Works

The script performs the following steps:

a. Initial Guess

var x=guess;

The Newton method needs a starting value.

b. Iterative Calculation

for(var i=0;i<iterations;i=i+1){...}

The algorithm runs up to 1000 iterations to find a solution.

c. Evaluate Function and Derivative

var f=x*ln(x)-ln(k);

var df=ln(x)+1;

These correspond directly to the mathematical formulas derived earlier.

d. Newton Update

var nextX=x-f/df;

This step improves the estimate of the solution.

e. Convergence Check

if(abs(nextX-x)<tolerance){...}

When the change becomes extremely small, the algorithm stops.

5. Example Usage

Suppose we want to solve

We run the function with an initial guess:

xPOWx(10,2)

2.506184...

Checking:

So the function successfully finds the solution.

6. Why Programmable Calculators Are Powerful

With programmable scripting you can:

• create custom equation solvers

• implement numerical algorithms

• automate repeated calculations

• build a personal math toolkit

Instead of being limited to built-in functions, you can extend the calculator to solve almost any problem.

Conclusion

In this tutorial we created a custom solver for the equation using the Newton method.

This example shows how programmable calculators can be used not only for calculations, but also for building numerical algorithms directly inside the calculator.

If you enjoy experimenting with math and programming, try writing your own solver functions and expand your calculator’s capabilities.