Adding Logs with the Same Base

Remember that logarithms are just exponents. For example, we know that $\log_{2}{8}=3$, where 3 is the exponent. This means that when we add logarithms, we are essentially adding exponents.

Now when have we added exponents before? That's right, when we're multiplying powers with the same base. For example, if I wanted to do $2^{3}\times2^{5}$, I would add exponents to get $2^{3+5}=2^{8}$. Expressing this as logarithms,

$\log_{2}{2^3\cdot2^5}=\log_{2}{2^{3+5}}=3+5=\log_{2}{2^3}+\log_{2}{2^5}$

More generally,

$\log_{x}{m}+\log_{x}{n}=\log_{x}{mn}$

We can also work backwards to arrive at the same conclusion:

$\log_{x}{(m\times n)}\\=\log_{x}{(x^{\log_{x}{m}}\times x^{\log_{x}{n}})}\\=\log_{x}{x^{(\log_{x}{m}+{\log_{x}}{n})}}\\=\log_{x}{m}+\log_{x}{n}$

Okay, so, we're given two numbers: 1 and 2.

Awesome! Now let's write the numbers 1 and 2 in log base 10. Remember, log base 10 is the standard base, so we can just write $\log(x)$, meaning "log base 10 of x."

Note that I'm assuming you know what logarithms are for this explanation. If not, you should find out about them from a previous explanation before coming back to this one.

So, as we saw previously, 1+2=3. Using log representation, therefore, log 10 + log 100 = 3.

That's cool. As it turns out, 3 = log 1000. There's something interesting here, what do you think it is?

log $10^1$ + log $10^2$ = log $10^3$

Look at the exponents, don't they look familiar? Quite so. In other words,

log $10^1$ + log $10^2$ = log $10^1 * 10^2$ = log $10^{1+2}$ = log $10^3$.

Isn't that awesome? Quite so, but as mathematicians, we like to generalize. I arbitrarily picked log base 10 because it's common, but I could've changed that. Also, I could've used this technique to add any numbers, not just 1 and 2. Therefore, I'll end this post with the generalized form and then we'll all be happy :-)

$\log_b m$ + $\log_b n$ = $\log_b mn$