Solving $2^n = 10^6$ for $n$

Question

I am trying to solve $2^n = 10^6$ for $n$. Is there some way that I can solve this analytically (say, using base $2$ logarithms), or must I solve it numerically? Thank you.

Answer 1

Take the logarithm base $2$ of both sides and use properties of logarithm to get $n = \log_2 10^6=6 \log_2 10$.

Answer 2

As a minor alternate to the answer of @J.W. Tanner:

Taking logarithms to the base 10 of both sides: $$2^n=10^6$$ $$\log(2^n)=\log(10^6)$$ $$n\log(2)=6$$ $$n=\frac{6}{\log(2)}$$Common logarithm are slightly more accessible than logarithms base $2$

Answer 3

Well, sure. $\log_2 2^n = \log_2 10^6$

$n = 6\log_2 10$ or

$\log_{10} 2^n = \log_{10} 10^6$

$n \log_{10} = 6$

$n = \frac {6}{\log_{10}2}$ or

$\log_b 2^n = \log_b 10^6$

$n\log_b 2^n = 6\log_b 10$

$n = 6\frac {\log_b 10}{\log_b 2}$ for any $b > 0; b\ne 1$ are all acceptable answers.

But it's worth noting that

1) $\frac {\log_b 10}{\log_b 2} = \log_2 {10}$ always. that's called the change of base formula and a special case is $\frac 1{\log_{10} 2}=\frac {\log_{10} 10}{\log_{10} 2} = \log_2 {10}$.

2) $\log_2 10= \log_2 (2*5) = \log_2 2 + \log_2 5 = 1 + \log_2 5$.

So $2^n = 10^6 = 2^6*5^6$

$\log_2 2^n = \log_2 (2^6*5^6)$

$n = \log_2 2^6 + \log_2 5^6 = 6 + 6\log_2 5=6(1 + \log_2 5)$.

All acceptable answers.