Intuition for high school students regarding square roots and logarithms

Question

These are some common mistakes high schoolers make:

$$ \sqrt{a + b} = \sqrt{a} + \sqrt{b} $$

$$ \log(a+b) = \log (a) + \log(b)$$

So I can obviously show numeric examples to say why these are wrong, but I want to show why in general these are wrong. What are some intuitive arguments to show these are wrong? For example, for $(a+b)^2$ there are some nice visual (geometric) illustrations which show why it equals $a^2 + b^2 + 2ab$, and I'd like some similar examples for the more difficult square roots and logarithms.

Answer 1

It is easier to show this for the inverse functions $x \mapsto x^2$ and $t \mapsto e^t$ respectively, where you can use known rules of binomial experssions and power laws.

E.g. $$a + b = e^{\log(a+b)} \ne e^{\log a + \log b} = e^{\log a} e^{\log b} = ab \ .$$

Answer 2

For the roots, you could choose some Pythagorean triples $a,b,c$ (i.e. such that $a^2+b^2=c^2$) and make them draw the squares of sides $a,b,c$ and then think of what happens to the area of the squares. For instance, if $a=3, b=4$ and $c=5$ you have three squares of area $9,16$ and $25$. The areas of the smaller squares add up to the big area, but not the sides (square roots).

As for the $\log$, maybe a way to go is talking about the "integer" version of the function defined on powers of $10$, that is $\log_{10}(x)$. This function counts the zeroes of a number. If you have, say, the number $100000$, the result is $5$. Make them reflect on the number $5$ rather than on the original number. Make them write $5$ as a sum, for instance $2+3$, and work on what that means $(100000 = 100\cdot1000)$...maybe they grow to like playing with these kind of things!

Answer 3

For the first one, why don't you ask to square on both sides - which will give $a+b = a+b+2\sqrt{ab}$, which implies at least one of $a$ and $b$ have to be zero. One can try and come up with similar arguments for the second one, by using $\log (a) + \log (b) = \log(a+b)$ (the only real solution would be a=b=2 in this case).