Question

question

Answer 1

\(\large \color{black}{\begin{align} \text{find }\ \log_{10}3\ \text{without calculator }\ \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

@SithsAndGiggles

Answer 3

It really depends on how accurate you'd like it to be... We can use the power series of \(\ln(1+x)\) to make a decent approximation.
\[\ln(1+x)=-\sum_{n=1}^\infty\frac{(-x)^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots\]
Change the base to the natural one, then use the series above:
\[\log_{10}3=\frac{\ln3}{\ln10}=\frac{\ln(1+2)}{\ln(1+9)}\]
At least that's how I would approach it if I didn't have the luxury of using a calculator.

Answer 4

i need 2 decimal places

Answer 5

how long do need to write the first series for 2 decimal places

Answer 6

approximation correct upto 2 decimal places