]> Exercise 2.14

## 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.

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