IB HL Math Proof: Log Base Change Solution
Assigned: 8/24/07, Friday
Due: 8/2907, Wednesday
A technique shown in class to solve problems that involve logarithms was to change the
base of a logarithm via the following identity:
log a b 
log c b
log c a
Utilizing the technique of introducing new variables and using exponent rules (which we
have already proven) to prove the identity shown above.
Solution
Let x  log a b . Thus, be definition, we have that a x  b . Now, take log base c of both
sides of this equation:
log c a x  log c b , now, use the power rule previously proven in class to yield:
x log c a  log c b , divide...
log c b
, now, just substitute back for x to prove the result:
log c a
log c b
log a b 
log c a
x
Note: A different proof is show in the log notes posted online.