How do you convert a logarithm to a different base?

1 Answer
Oct 22, 2015

Answer:

If you want to convert #log_a(x)# into #log_b(x)#, use the following equality:
#log_b(x) = log_b(a)*log_a(x)#

Explanation:

Consider #y=log_a(x)#.
From a definition of logarithm
(1) #x = a^y#
Analogously, if #z=log_b(x)#,
(2) #x = b^z#
Finally, if #r=log_b(a)#,
(3) #a = b^r#

From (1) and (2) we derive
(4) #a^y = b^z#

Using (3) and substituting #a# in (4) we get
(5) #(b^r)^y = b^z#
or, simplifying the left part,
(6) #b^(r*y) = b^z#

From the last equation we see that
#z = r*y#
or, returning back to the meaning of #r#, #y# and #z#,
#log_b(x)=log_b(a)*log_a(x)#