Apply algebraic reasoning

Question

Show that a=b^(loga/logb) for any a,b>0.

Answer 1

Here I assume that log(x) stands for log of base 10 but the argument is valid if it means log of any other fixed base (e.g., e).

 

The key is to know that 1/log(x) = 1/log_10(x) = log_x(10), where log_c means log of base c.

Then b^(log(a)/log(b)) = [ b^(1/log(b)) ]^log(a) = [ b^(log_b(10)) ]^log(a) = 10^log(a) = 10^log_10(a) = a,

where for the last and third to last equalities I used the property x = y^log_y(x) valid for any x and y.