Is $\log(100 \cdot 10^x)$ the same as $\log_{10}(100 \cdot 10^x)$?

Question

I just learned about logarithms, and my question is:

Is $\log(100 \cdot 10^x)$ the same as $\log_{10}(100 \cdot 10^x)$?

If so, why?

Answer 1

In general this is a little bit tricky. I have seen three important cases concerning the base conventions of logarithms

$$\begin{align} &\text{The natural logarithm }&&\log_e(x)\text{ often denoted as }\ln(x)\text{ but also sometimes as }\log(x)\\ &\text{The decadic logarithm }&&\log_{10}(x)\text{ often denoted as }\lg(x)\text{ but also sometimes as }\log(x)\\ &\text{The binary logarithm }&&\log_2(x)\text{ often denoted as }\operatorname{ld}(x)\text{ or }\operatorname{lb}(x)\\ \end{align}$$

So it is a matter of context I would say but not a general fact that $\log(x)$ refers to the decadic one. I for myself tend to use $\log(x)$ for the natural logarithm.

Answer 2

depending where you are from, in some areas $\log(x)=\log_{10}(x)$ and $\ln(x)=log_{e}(x)$ whilst in other areas people take $\log(x)=\log_{e}(x)$ so it is best to use $\log_{10}(x)$ and $\ln(x)$ to make it clear.