Did you know you can calculate logarithms with a basic calculator?
Discover a unique method to calculate logarithms using basic calculators through iterative calculation and Taylor Series Expansion.
During my high-school days, most students in India couldn't afford a decent calculator, resorting to cheaper models capable only of basic BODMAS operations and square roots.
Log books were scarce, often going out of stock quickly.
However, there was a remarkable trick to find the log₁₀ of any number using this regular calculator:
The Remarkable Trick
Type the number.
Take its square root 19 times, recursively.
Subtract one.
Multiply the result by 227697.
This simple process yielded the log₁₀(x) accurate up to 4/5 decimal places or even more.
Reverse Engineering the Trick
Reverse engineering the method, the explanation behind this trick can be unveiled.
The Mathematics Behind It
The trick exploits a mathematical expression known as the Taylor Series Expansion of the logarithm function.
The Taylor Series provide an approximation of a function around a given point.
With this trick, the repeatedly taking the square root represents an iterative method to approach the logarithm function value.
The subtracting and multiplication steps contribute to refining the approximation to yield the log₁₀(x) value.
The Role of Convergence
The effectiveness of the trick lies in the convergence property of the series.
The algorithm stabilizes and approaches a unique value after a sufficient number of iterations, yielding an accurate result.
Numerical Stability
The stability of this method is also a critical factor.
Contrary to standard calculational limitations, this iterative process advances the accuracy of the result through round-off and truncation errors.
In essence, by leveraging fundamental mathematical concepts and properties, this simple yet clever trick has enabled students with limited resources to obtain accurate logarithmic values, showcasing the power of mathematical ingenuity.
Conclusion
The poor man's log₁₀ calculator trick is a testament to the resourcefulness and creativity born from necessity, exemplifying the practical application of mathematical concepts to navigate everyday challenges.
