How to convert from a power of base two to a power of base 10?

I might have an extremely silly question: If I have a number, say $2^{32}$ and I need to convert to base 10, how should I do it? I know it should be $4 * 10^9$, but I do not know how did we get it.

I understand that $10$ is $2^3 + 2$, but I cannot understand how to proceed further in my reasoning...

Thanks!

Solution 1:

$$ \log_{10} (2^{32}) = \frac{32 \log 2}{\log 10} \approx 9.633 $$ and since $10^{0.633} \approx 4$, $$ 2^{32} \approx 4 \cdot 10^9$$. Or, notice that $2^{10} \approx 10^3$. It follows that $2^{30} \approx 10^9$, and $2^{32} \approx 4 \cdot 10^9$.

Solution 2:

To brush up on algebra rules for exponents, the relevant rule here is that $x^{A*B}=(x^A)^B$. And as the others have already mentioned, $2^{10}=1024≈10^3=1000$. If we pick an example number that is so horrendously gigantic where this small inaccuracy doesn't matter, something like $2^{(2^{20})}=2^{1,048,576}$, you can factor out the exponent and replace it as follows: $$2^{1,048,576}=2^{10*104,857}=(2^{10})^{104,857}=(10^3)^{104,857}=10^{3*104,857}=10^{314,571}$$ Extremely simple, as long as you remember the algebra fundamentals :)

Solution 3:

To work with exponents of different bases, you also need its opposite operation, the logarithm:

$$ 2^x = 10^y $$ applying a $log_{10}$ to the previous equation:

$$ y = \log_{10} 2^x = x·\log_{10}2 = 0.30103x $$

Therefore (in your question; $2^{32}$):

$$ y = 32×0.30103 = 9.63296 $$ $$ 2^{32} \sim 10^{9.6} $$

Summary: The factor $0.3$ can be used to multiply the exponent of $2$ to obtain the exponent of $10$, or can be used to divide the exponent of $10$ to get the exponent of $2$, and the approximation is very high, particularly for big exponents.