why does $2^{\log_2c} = c$

Question

Reviewing some basic properties log logs. why does $2^{\log_2c} = c$? I understand that $\log_2c$ is basically $2$ to some power = $c$, but I'm not sure why the above expression is equal to $c$. thanks!

Answer 1

$\log_2 c$ is defined as the unique number $u$ such that $2^u = c.$ Therefore $2^{\log_2 c} = c.$

Answer 2

$\log _2 c = a$ for some $a$ if $2^a =c$. Hence $2^a =c$ and $2^{\log_2 c}=c$.

Answer 3

For any $b>0$, $\log_b x$ is the inverse function of the function $b^x$, and, as is the case for all inverse functions, $$f\circ f^{-1}=\operatorname{id}, \enspace\text{so }\quad b^{\log_b(x)}=x\enspace\text{and}\enspace\log_b(b^x)=x.$$

Answer 4

The exponential and logarithmic functions have an inverse relation; that is to say, $$f(x)=b^x\iff f^{-1}(x)=\log_b(x).$$ As such, remembering what you know about functions and their inverses, it is easy to see why you have what you have: $$(f\circ f^{-1})(x)=x$$

Let $f(c)=2^c$. Therefore, $f^{-1}(c)=\log_2c$, and $2^{\log_2c}=c$.

Answer 5

By definition $k= \log_2 c$ is the solution to $2^k = c$.

So if $2^{\log_2 c} \ne 2$ then $k= \log_2 c$ is NOT the solution to $2^k = c$.