0
$\begingroup$

I am going back to study log and unfortunately I don't know a lot. I need to solve this: $$ 100= 10\log_{10} \left(50/x\right) $$ I did the wrong calculation just moving stuff to the left, but I've been told is not right: $$\begin{align} 100 \cdot x &= 10\log_{10} \cdot 50\\ x &= \left(10\log_{10} \cdot 50\right) / 100\\ x &= 0.016 \end{align}$$ I know that it is wrong, can someone explain me how to solve this?

$\endgroup$
  • $\begingroup$ Could you use brackets in your formulas? It's not clear if you mean $^{10}\log(10)\cdot\frac{50}x$, $^{10}\log\left(10\cdot\frac{50}x\right)$ or $10\log_{10}\left(\frac{50}x\right)$. I suppose it's the third one. $\endgroup$ – punctured dusk Mar 4 '14 at 9:57
  • $\begingroup$ yes it's the third one barto, thanks a lot, sorry I am learning to use the site $\endgroup$ – jsab Mar 4 '14 at 9:59
5
$\begingroup$

$x\ne 0$ is part of the argument of $\log_{10},\,$ you cannot move it out the way you did. You can use $\log_{10}(50/x)= \log_{10}50 -\log_{10}x,\,$ or something like this: $$100 = 10\log_{10}\left(\frac{50}{x}\right)\quad (x\ne0)$$ $$\iff 10 = \log_{10}\left(\frac{50}{x}\right)$$ $$\iff 10^{10}=\frac{50}{x}$$ $$\iff x=\frac{50}{10^{10}}=\frac{1}{200000000}=5\times10^{-9}$$

$\endgroup$
2
$\begingroup$

The mistake is that we can't just pull the factor $\frac1x$ out of the logarithm, i.e, $$\log_{10}\left(\frac{50}x\right)\neq\log_{10}(50)\cdot\frac1x$$ in general. The logarithm (in particular the base-$10$ logarithm) has no rule like $$\log(a\cdot b)=\log(a)\cdot\log(b).$$ (In fact we have $\log ab=\log a+\log b$.)

What we can use here is: $$\log_{10}(a)=b\quad\Longleftrightarrow\quad a=10^b.$$

Before making this possible, we need a small manipulation to the given equation: $$100=10\cdot\log_{10}\left(\frac{50}x\right)\quad\Longleftrightarrow\quad10=\log_{10}\left(\frac{50}x\right).$$ Now we can apply the above rule, with $a=\frac{50}x$ and $b=10$: $$\frac{50}x=a=10^b=10^{10}.$$ Look, the logarithm has disappeared! From here we can conclude $$x=\frac{50}{10^{10}}=\frac1{200000000}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.