0

When someone says "8 Log 2" what does this equate to in writing?

Does it mean the following?

$$ \log _{2} 8 $$

And if so, what is the value of this?

When I plug those numbers into this log calculator, it tells me 3.

But when I plug it into Google's calculator, it tells me 2.4082

Saqib Ali
  • 125
  • 4
    What you put in the calculator is interpreted as $8*log(2)$, and Google uses the base 10 logarithm. $log_28$ would be read as "log base 2 of 8." – alphacapture May 26 '16 at 02:23
  • 8
    Honestly, it depends, but it never should mean $\log_2 8$. It means 8 times the logarithm of $2$, but, without a context, that might mean log base $10$, the natural logarithm, or log base $2$. Log base $2$ is unlikely in this case, since $\log_2 2$ is trivial. – Thomas Andrews May 26 '16 at 02:25
  • 2
    The standard way of reading $\log_{a}b$ is "log base $a$ of $b$". – Pockets May 26 '16 at 04:59

7 Answers7

6

Google's calculator is treating it as $8\times \log_{10} 2$ and the "log calculator" to which you link is finding $\log_2 8$.

From the fact that $2^3=8$, it follows that $\log_2 8=3$.

$8\log 2$ means $8\log_b 2$, where $b$ is some positive number other than $1$. The expression does not specify which number $b$ is. In some contexts in science and engineering, it is conventional that when $b$ is unspecified, then $b=10$. In theoretical mathematics, it is usually taken to be $e$.

Michael Hardy
  • 1
5

I would think $8$ log $2$ means the person is usually looking for $8\log 2$

Even so, It's not very clear base you are in. Usually if you are in a science class Log is specifically referring the logarithm of a number base $10$. But mathematicians like to use Log to mean the logarithm base $e$.

What you written is wrong, but let's just do the math behind it:

$$\log_2 (8)= \log_2 (2^3)$$

First let me note that note that $\log_a (b)$ is defined to be the solution of $b=a^x$ for $x$.

Now there is a rule that results from exponents that says,

$$\log_x (a^b) = b \log_x (a)$$

If we use this we get,

$$\log_2 (2^3) =3 \log_2 (2)$$

And

$$\log_x (x)=1$$

Because the $r$ that will make $x^r=x$ true for all $x$ is 1.

So like you said it is $3$ if $2$ is the base.

Ahmed S. Attaalla
  • 18,518
3

If someone says "eight log $2$" they mean this:

$$8\log2$$

Where $\log=\log_{10}$ and is interchanged with the terms "common log" or "log base $10$". It can be rewritten as

$$8\log_{10}2$$

Edit: Though from my experience it almost always refers to base $10$, judging by the comments below this post "eight log 2" could also be rewritten as the following as well:

$$8\ln2$$

$$8\log_22$$

Essentially, "eight log two" is an ambiguous phrase and needs more context.

What you've written, $\log_28$, is read as "log of $8$ to base $2$".

Leonidas Lanier
  • 4,149
  • 7
    I agree with the first part, but $\log 2$ might mean base $e$ or base $2$, depending on who is speaking. – vadim123 May 26 '16 at 02:24
  • 3
    I never hear "log to the base $10$". I do hear "log base $10$". "To the" to me implies a superscript. –  May 26 '16 at 02:24
  • I have always been taught that $\log$ is for base ten and $\ln$ is for base $e$.

    Under what circumstances have you seen it mean differently?

     – Leonidas Lanier May 26 '16 at 02:25
  • @ZacharySelk I have heard both used interchangeably. – Leonidas Lanier May 26 '16 at 02:26
  • 2
    Given the apparent level of this question, I think we can be fairly certain it's base ten. – jdods May 26 '16 at 02:42
  • 5
    @LanierFreeman in more advanced math it's typical to use $\log$ to refer to the natural logarithm. $\ln$ is still also used but I've more frequently seen $\log$ in more advanced (relative to high school etc.) material. For example, see almost any book on complex analysis, partial differential equations, etc. I've also seen it in a few calculus textbooks but they're either from outside the US or are from the US and decades old. –  May 26 '16 at 03:32
  • @tilper Thank you for the explanation, this probably explains why I've not seen it used as much. – Leonidas Lanier May 26 '16 at 03:36
2

I would choose to clarify with the person what does he mean.

I could have interpreted it as $8\log_{10}2$ or $8 \ln2$ or $8\log_22$.

Siong Thye Goh
  • 149,520
  • 20
  • 88
  • 149
1

First of all you must know that whenever the base of a log is not mentioned then its base is 10.

$$m\log a= \log(a^m)$$

It is wrong to assume it means "$\log 8$ to the base $2$".

The correct value will be $2.4082…$ not $3$. You might get a value of 3 in that calculator because the base was not set to 10 or due to some other mistake.

grg
  • 1,017
JM97
  • 292
1

I believe if some says "8 Log2" just means the logarithm of 8 * Log(2), otherwise if he intents to specify the base as 2 he should say "8's Log in base 2" that actually is 3.

albanx
  • 111
1

8log 2 means , find [log (base 10) 2] and then multiply the answer with 8.

Amruth A
  • 167