## mathmath333 one year ago question

1. mathmath333

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

2. mathmath333

@SithsAndGiggles

3. anonymous

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.

4. mathmath333

i need 2 decimal places

5. mathmath333

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

6. mathmath333

approximation correct upto 2 decimal places

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