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A wiki page says

In mathematics, a base or radix is the number of different digits or combination of digits and letters that a system of counting uses to represent numbers. For example, the most common base used today is the decimal system. Because "dec" means 10, it uses the 10 digits from 0 to 9. Most people think that we most often use base 10 because we have 10 fingers.

Are the "base" here and the "base" in logarithm the same concept?

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Yes, they are closely related. In both cases it's about the base of the powers we use to describe a number. For number systems, saying we are using base ten means we express every number as a linear combination of integral powers of ten. For instance: $$ 345=3\cdot10^2+4\cdot10^1+5\cdot10^0 $$ This should be well-known from elementary school, except, perhaps, the usage of exponents.

When we take logarithms base ten, we express a number as a single power of ten. For instance: $$ 345=10^{\log345}=10^{2.537819\ldots} $$ In both cases we see $10^2$, since $345$ is between $100$ and $1000$. But in the logarithm case, we take the $3$ in front of the $10^2$, and all the smaller terms, and "absorb" them into the exponent.

One difference between the two is that making sense of a base that is not an integer is easier for the logarithm than for the number system. We routinely do logarithms with base $e$, yet it's not entirely obvious that a base $e$ number system would even work.

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  • $\begingroup$ Thanks for your answer. I am aware of "power of a number", for instance, three to the power of four, $3^4 = 81$; I am also aware of Integration, for example, ∫a dx = ax + C. What does "integral powers" mean? $\endgroup$ – JJJohn Feb 15 at 8:43
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    $\begingroup$ @JJJohn It means that the exponents are integers. $3^4$ is an integral power of $3$, while $3^{4.5}$ is not. $\endgroup$ – Arthur Feb 15 at 8:56
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No, they are totally different concepts. The base in number system tell us how many digits (or rather symbols) we have available to count. This gives us a one-one correspondence between the number of amount and its representation. In computers, bases $2$(binary) and $16$(hexadecimal) are very common. By default, we always use base $10$ (why we do that is explained here). Every number can be represented as the following- $$n=\sum_{i=0}^ka_ib^i\;\;\;\;\text{ (where $b$ is base and $a_i<b$)}$$ The base in logarithm tells us the number whose power gives another number. For example, $a^b=c$ implies that $\log_ac=b$. This means that when we raise $a$ to some power $b$, we get $c$ as a result. However, when used in math you should assume $\log x$ to refer to $\log_ex$ which may also be written as $\ln x$.

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  • $\begingroup$ Oh but I've been taught to assume that the base is 10 for $\log x$, instead of base $e$. That's even what my calculator uses.(The log button is base 10, with a separate button for natural log.). Is this way of thinking also correct or is it generally correct to assume $\log x$ is $\log_e x$? $\endgroup$ – Aiden Chow Feb 15 at 6:58
  • $\begingroup$ My calculator is the Texas Instrument TI-30Xa btw. $\endgroup$ – Aiden Chow Feb 15 at 6:59
  • $\begingroup$ It usually is base $10$ however, if the formulae involves any sort of calculus it will be base $e$. $\endgroup$ – Sam Feb 15 at 9:47

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