Home  18.013A  Chapter 2  Section 2.4 


Deduce from these two equations that for $\mathrm{log}$ to any base we have $\mathrm{log}ab=\mathrm{log}a+\mathrm{log}b$ .
Solution:
Suppose we want to prove this for logarithms to base $c$ . We deduced in the previous exercise that we can write ${\mathrm{log}}_{c}x=\frac{\mathrm{ln}x}{\mathrm{ln}c}$ .
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 $\mathrm{ln}c$ .
In other words, the corresponding result for natural logarithms, which we proved in the previous exercise, implies this one on dividing each term by $\mathrm{ln}c$ , and using the fact just mentioned.
