Proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.
Change all to the natural logarithm $\log\,$:

$$\log_ba\cdot\log_cb\cdot\log_ac=\frac{\log a}{\log b}\frac{\log b}{\log c}\frac{\log c}{\log a}$$

and voila.
Before we prove the given identity proof this idenity

$$\log_b a\log_c b=\log_c a$$

Proof: Implement the formula $\log_a b=\frac{\log_x b}{\log_x a}$

$$\frac{\log a}{\log b}\cdot\frac{\log b}{\log c}=\frac{\log a}{\log c}=\log_c a$$

Now proof the given identity.

$$\log_b a\cdot\log_c b\cdot\log_a c=1$$

$$\log_c a\cdot\log_a c=1$$

$$\frac{1}{\log_a c}\cdot\log_a c=1$$

$$1=1$$