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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

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7 Answers

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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 $e$.

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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.

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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 $2$".

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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 $8\log_22$.

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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.

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

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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.

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