I am reading 'A beginners guide to discrete mathematics'. And i stumbled on this proof they have:
Theorem. For any x and any base b, $x = log_{b}(b^{x})$
Proof. By definition, $u = b\text{ }log_{b}(u)$. Putting $ u = b^{x}$ we have $bx = b(log_{b}(b^{x}))$. So comparing the exponents, $x = log_{b}(b^{x})$
How the book writes it:
Why does $u = b\text{ }log_{b}(u)$ make any sense? e.g. $16 = 2*log_{2}(16)$, so what they are saying is that $16 = 8$ ? or is there something i am misunderstanding by the way they write "By definition" ?
This appear to be a typo. Try $u = b^{\log_b u}$, which is true (and differs mostly in white space). The same typographical error comes when setting $u =b^x$, but "$bx$" appears on the left-hand side of the substituted equation.