# Are the “base” in number system and the “base” in logarithm the same concept?

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?

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.

• 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? – JJJohn Feb 15 at 8:43
• @JJJohn It means that the exponents are integers. $3^4$ is an integral power of $3$, while $3^{4.5}$ is not. – Arthur Feb 15 at 8:56

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- $$n=\sum_{i=0}^ka_ib^i\;\;\;\;\text{ (where b is base and a_i The base in logarithm tells us the number whose power gives another number. For example, $$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$$.

• 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 $\log_e x$? – Aiden Chow Feb 15 at 6:58
• My calculator is the Texas Instrument TI-30Xa btw. – Aiden Chow Feb 15 at 6:59
• It usually is base $10$ however, if the formulae involves any sort of calculus it will be base $e$. – Sam Feb 15 at 9:47