Logarithmus to simple subtraction - how?

I am learning for a math exam and have the following solution:

$$0.01 = 0.5^n\\ n \cdot \log 0.5 = \log 0.01\\ n=\frac{\log 0.01}{\log 0.5}$$

OK, so far, so good. (I guess)

But now, it gets weird:

$$n=\frac{\log 0.01}{\log 0.5}=\frac{0-2}{0.7-1}=…$$

Can somebody please explain how to go from $\log 0.01$ to $0-2$ and from $\log 0.5$ to $0.7-1$?

For the numerator, notice that $.01 = \frac{1}{100}$ and by logarithmic rules $$\log .01=\log\frac{1}{100}=\log1-\log100=0-2$$ The same goes for the denominator: $.5=\frac{5}{10}$ $$\log .5=\log \frac{5}{10}=\log5-\log10=0.7-1$$

Using the logarithmic rule: $$\log_a\left(\frac{m}{n}\right)=\log_a m-\log_an$$

• Nice, thanks. How would I know that log 5 = 0.7? The 0.7 is not even exact, right? Feb 15, 2014 at 14:23
• Are you allowed to use calculators? You'd get $\log 5=0.69897...\approx .7$.
– Zhoe
Feb 15, 2014 at 14:24
• Yeah, I am allowed. But with a calculator, I could just punch in $\log 0.5$ directly... Feb 15, 2014 at 14:25
• I would have done the same..I am not sure why the solution bothers to expand, but you'll get the same answer.
– Zhoe
Feb 15, 2014 at 14:27
• OK, thanks! :-) Feb 15, 2014 at 14:57

log0.01 = log 1/100 and log 0.5 = log(5/10) now apply log(m/n) = log m -log n

log 0.01 = log(1/100) = log 1 - log 100 = log 1 - 2 log 10 = 0-2

log 0.5 = log(2/10) = log 2 - log 10 = 0.7 - 1

log(a/b) = log a -log b

log(a^n) = n log a

Hope this helps!

$$n=\frac{log0.01}{log0.5}$$.Remebner that $logm^n=mlogn$.Therefore $log 0.01=log$ $\mathrm{10}^{-2}$=$-2$.$log.5=log \frac{1}{2}=log1-log2=0-log2$

$$n \ = \ \frac{log_{10} \frac{1}{100}}{ log_{10} \frac{5}{10}}$$

Can you finish it now?

As mentioned above, it comes from the properties of quotients of logs. It looks like the way logs were written when people looked them up in tables (the tables went from $\log 1$ to $\log 9.999$, and you added/subtracted the exponent (as if you had written the number in scientific notation)), but I have no idea if they're actually expecting you to use tables here.