How many Digits in $3^{2020}$

Question

I know n+1 = number of digits in $3^{2020}$

but how to find number of digits .

How can we solve it without using a calculator .

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can we not solve it using binomial

$3^{2020}=9^{1010}=(10-1)^{1010}$

after that I am stuck

Answer 1

Let $x=3^{2020}$

$$\lg x = 2020 \lg 3 \approx 2020(0.477)=963.7$$

$$x=10^{963.7}$$

Hence there are $964$ digits.

Answer 2

$3^{9} =19683 \approx 20000 \implies 9 \log 3 \approx 4 + \log 2$

$2^{10} = 1024 \approx 1000 \implies \log 2 \approx 0.3$

$ \implies \log 3 \approx \dfrac{4.3}{9} \approx 0.477$

$ \implies \log 3^{2020} = 20020 \log 3 \approx 963.54$

So $3^{2020}$ has $964$ digits.

All this can be made precise by using inequalities.