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### The computer can't do anything I couldn't do with pen and paper

An elementary approach:
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### The computer can't do anything I couldn't do with pen and paper

Not sure if this counts as "pencil-and-paper" but certainly counts as "no computer..." $$3^{100} = 10^{100 \log 3}$$ We can compute this logarithm using a slide rule. The D and L ...
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### The computer can't do anything I couldn't do with pen and paper

Let's solve this problem using ... music! Because of this, But also,
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### The computer can't do anything I couldn't do with pen and paper

I took an extra challenge to solve this in my head without writing. From here, we can use that Now which is quite close to the true value.
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### The computer can't do anything I couldn't do with pen and paper

This seems like a perfect application for Looking at Now for the inverse operation: Bringing it all together, Error analysis:
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### The computer can't do anything I couldn't do with pen and paper

Simple approach... I don't find a simple way to quantify the error:

### How many digits does $7^{100}$ have?

Once again, I solved this in my head, without writing or looking anything up. We use the approximation and also rewrite and estimate to get which is quite close and gives the right digit count of
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### The computer can't do anything I couldn't do with pen and paper

Second answer, using exponentiation by squaring and one (!) digit of precision:  \begin{align} 3^{1} &= 3 \\ 3^{2} &= 3^{2} = 9 \\ 3^{4} &= 9^{2} = 81 \approx 8 \times 10^{1} \\ 3^{8} &...
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### The computer can't do anything I couldn't do with pen and paper

Note first of all that Well, Now At this point The errors here are at most about so comfortably within the requested factor of 2.
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Pretty similar to other solutions: $3^{10} = 9^5$ $9^n = 9, 81, 729, 6561, 59048$. (Can be done on paper, or even in the head) $3^{10} = 59048 \approx 6 \times 10^4$ $3^{100} \approx 6^{10} \times 10^... • 22.7k 5 votes ### The computer can't do anything I couldn't do with pen and paper I already posted one approach. Several others are possible. Here's one that gives a pretty accurate result. It's similar in spirit to my other answer. Now So We can already but • 112k 4 votes ### Can you Avoid the Spear-Wielding Gladiator? • 4,113 3 votes ### The computer can't do anything I couldn't do with pen and paper Another 80 based solution. • 5,716 3 votes ### The computer can't do anything I couldn't do with pen and paper Short and rough:$3^5=243\approx 250 = 10^3/2^2$So$3^{100}\approx (10^3/2^2)^{20} = 10^{60}/2^{40}$But$2^{10}=1024\approx 10^3$So$2^{40}\approx 10^{12}$Hence$3^{100}\approx 10^{48}$• 195 3 votes ### How many digits does$7^{100}\$ have?

A simple upper bound: And a semi-simple lower bound: Or, borrowing an answer from the previous problem: If you have then you can determine that which produces an estimate of
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