Mathconceptions
=============

# Correct results from **bogus** rules

## Rules of logarithms

Incorrect rules such as

* $\log(a) + \log(b) = \color{red}{ \log(a)\cdot\log(b)=}\log(a\cdot b)$
* $b\cdot \log(a) = \color{red}{\log(a)^b=}\log(a^b)$

Here is an example from student work:
$$
\log(11)-\log(x)+3\log(\sqrt{x})=\color{red}{\frac{\log(11)\cdot\log(x^{1/2})^3}{\log(x)}}=\log\left(\frac{11\cdot x^{3/2}}{x}\right)=\log(11\cdot x^{1/2})
$$
Of course removing the $\color{red}{\text{incorrect}}$ middle expression from each of the two previous rules results in something true, so it is not surprising that true results are deduced from these bogus rules.

## Multiplication of binomials

$$(a+b)(a-b) =\color{red}{ (a)\cdot(a) + (b)\cdot(-b)} = a^2-b^2$$

## Solving inequalities
Solve $\frac{x-2}{-2}<3 - x$:
$$
\begin{align*}
\frac{x-2}{-2}&<3 - x\\
(-2)\left(\frac{x-2}{-2}\right)&\color{red}{<}(-2)(3 - x)\tag*{(Wrong)}\\
x-2 &< -6 + 2x\\
-3x & < -4\\
x \color{red}{<} -4/-3 &= 4/3 \tag*{(Wrong)}
\end{align*}
$$
But the answer $(-\infty, 4/3]$ is correct.