-2
$\begingroup$

I'm looking for the methodology to convert an exponent from base $2$ to base $1000$. If: $$2^{12000000000000} = 1000^x$$ What does $x$ equal?

  • 4
    $a^b=c^x\implies b=x\log_a c\implies x=\frac b{\log_a c}$ – 2017-02-14
  • 1
    Just take log base 1000, right? – 2017-02-14

2 Answers 2

1

The standard approach is take log base 1000 of both sides:

$$2^{12000000000000}=1000^x$$

$$x=\log_{1000}(2^{12000000000000})=12000000000000\log_{1000}(2)\\=4000000000000\log_{10}(2)\\\boxed{\approx1204119982655.92478}$$

Where we appropriately use log rules.

If you were asked to do this without calculators, and provide a rough approximation, then notice that

$$2^{10}=1024$$

Thus,

$$2^{12000000000000}=1024^{1200000000000}\approx1000^{1200000000000}$$

Thus, $x\approx1200000000000$, which is fairly decent of an approximation.

1

$$2^{12000000000000} = 1000^x$$ What does $x$ equal?

Then you want to express $2^{12000000000000}$ as a power of $1000$

Just take the logarithm of $2^{12000000000000}$ in base $1000$:

$x = \log_{1000}(2^{12000000000000}) = 12000000000000\cdot log_{1000}(2) \approx 1.204119982655925\cdot10^{12}$