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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?

Blue
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Hilmar
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  • It's the question of definition. Usually it's used $\lg$ instead of $\log_{10}$. And $\log$ as $\log_{e}$ – openspace Oct 28 '18 at 19:38
  • And in other contexts, $\lg$ often means $\log_2$ and $\log_e$ is often denoted by $\ln$. The bottom line is, when you see $\log$ without a base, think of whatever the base could be. – metamorphy Oct 28 '18 at 19:39
  • If I type the two logarithms in a math program, I get the same answer. – Hilmar Oct 28 '18 at 19:40
  • @Hilmar So within your math programm the short notation $\log(x)$ refers to the decadic logarithm. – mrtaurho Oct 28 '18 at 19:43

2 Answers2

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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.

mrtaurho
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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.

Henry Lee
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