Home | 18.013A | Chapter 2 | Section 2.4 | | Tools Glossary Index Up Previous Next

Exercise 2.14

Deduce from these two equations that for log ⁡ to any base we have log ⁡ a b = log ⁡ a + log ⁡ b .

Solution:

Suppose we want to prove this for logarithms to base c . We deduced in the previous exercise that we can write log ⁡ c x = ln ⁡ x ln ⁡ c .

Applying this fact to each logarithm here, the result we want to prove is the same result for natural logarithms, with each term divided by ln ⁡ c .

In other words, the corresponding result for natural logarithms, which we proved in the previous exercise, implies this one on dividing each term by ln ⁡ c , and using the fact just mentioned.

Up Previous Next