Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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. – barto Mar 4 '14 at 9:57
yes it's the third one barto, thanks a lot, sorry I am learning to use the site – jsab Mar 4 '14 at 9:59
up vote 5 down vote accepted

$x\ne 0$ 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: $$100 = 10\log_{10}\left(\frac{50}{x}\right)\quad (x\ne0)$$ $$\iff 10 = \log_{10}\left(\frac{50}{x}\right)$$ $$\iff 10^{10}=\frac{50}{x}$$ $$\iff x=\frac{50}{10^{10}}=\frac{1}{200000000}=5\times10^{-9}$$

share|cite|improve this answer

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 $$x=\frac{50}{10^{10}}=\frac1{200000000}.$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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