
log _{x} a = log _{y} a . log _{x} y. Proof: Let log _{x} a = p, Then x^{p} = a I log _{y} a = r, then y^{q} = a II also let log _{x} y = r, then y = x^{r} III From I and II we get, a = x^{p} = y^{q} IV From III, we get, y ^{q} = (x^{r})^{q} = x^{rq }V x^{p} =x^{rq} bases are equal and so indices are also equal. Therefore, p = rq log _{x} a = log _{y} a . log _{x} y. This result is used to change the base of the logarithm.
Note:
Example: Directions: Prove the following statements. Also write at least ten examples of your own. 1. Show that, log _{3} 2 . log _{4} 3 . log _{2} 4 = 1. 2. Show that, log _{b} a . log _{c} b . log _{a} c = 1. 3. Show that, log _{6} 5 . log _{7} 6 . log _{8} 7 . log_{5} 8 = 1.
4. Show that, log _{8} 9 . log _{7} 8 . log _{9} 7 = 1. 