Question

prove that logx divided by logy=logy basex

Answer 1

plez help

Answer 2

Are you sure you have stated the question correctly as I get a different result?

Using the change of base log rule, we know that:\[\log_b(x)=\frac{\log_d(x)}{\log_d(b)}\]so we can change \(\log_e\) to \(\log_x\) as follows:\[\log_e(x)=\frac{\log_x(x)}{\log_x(e)}=\frac{1}{\log_x(e)}\qquad\text{(as }\log_x(x)=1\text{)}\]similarly:\[\log_e(y)=\frac{\log_x(y)}{\log_x(e)}\]\[\therefore\log_e(x)\div\log_e(y)=\frac{1}{\log_x(e)}\div\frac{\log_x(y)}{\log_x(e)}=\frac{1}{\cancel{\log_x(e)}}\times\frac{\cancel{\log_x(e)}}{\log_x(y)}=\frac{1}{\log_x(y)}\]