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How many numbers between 100 and 200 can be expressed as powers of 2? Show the steps you followed.


Which power of 2 is closest to 1000?Show your work

Verified Answer :

superkoopasirf

Let x = the number 2 must be raised to to make 100
Let y = the number 2 must be raised to to make 200

\(2^x = 100\)
\(\log 2^x = \log 100\)
\(x \log 2 = \log{100}\)
\(x = \frac{\log 100}{\log 2}\)
\(x \approx 6.64\)

\(2^y = 200\)
\(\log 2^y = \log 200\)
\(y \log 2 = \log{200}\)
\(y = \frac{\log 200}{\log 2}\)
\(y \approx 7.64\)

So the integer powers of 2 that would fall between 100 and 200 are between 6.64 and 7.64 - 7 is the only one.

Following a similar method for the second question:
Let z = the number 2 must be raised to to make 1000

\(2^z = 1000\)
\(\log 2^z = \log 1000\)
\(z \log 2 = \log 1000\)
\(z = \frac{\log 1000}{\log 2}\)
\(z \approx 9.97\)

So \(2^{10}\) is the closest power of 2 to 1000.

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