# basic math question: transform a sum of exponents to a sum of logarithms

I am sure this is a really dumb question but I am having trouble understanding it since I do not have any math background.

I have the logarithms of 2 values:

log(a) = 1347
log(b) = 1351


I am trying to calculate this:

exp( log(a) ) - exp( log(0.1) + log(b) )


I guess by transforming this into a sum/difference of logarithms and working in log scale because when I exponentiate the values become infinite. Basically I am looking for a solution for this in log scale, so that I do not get an Infinite value when entered in a calculator.

Does this make sense?

 log(a)-(0.1*log(b))


Thank you!

-fra

-
That is just $a / (0.1 \cdot b)$ by basic properties of logaritms. Or am I overlooking something? – vonbrand Feb 3 '13 at 19:58
I do not have a because I cannot solve for it since exp(log(a)) = Infinite. That is why I was looking to transform everything in log form.. is this correct: log(a)-(0.1*log(b)) ? – Francesca Feb 4 '13 at 0:54

Hint: Recall that $$\exp(\log(a))=a\quad\text{and}\quad\exp(x+y)=\exp(x)\exp(y).$$ Here we're using the convention that the base of $\log$ is $e$.
Update: Using the first hint, we have $$\exp(\log a)=e^{1347}.$$ Combining the the second term of the equation, we have $$\frac{1}{10}\exp(\log b)=\frac{1}{10}\exp({1351}).$$Now, combining the expressions, $$\exp(\log a)-\exp(\log(1/10)+\log(b))=e^{1347}-\frac{1}{10}e^{1351}.$$
@Francesca: Since we know $\log(a)=1347$, we can deduce that $$a=\exp(1347).$$ If you try to put this in a calculator, it surely will throw an error, but you can use this. Does clarify anything for you? – Clayton Feb 3 '13 at 14:03
$@Francesca: I've updated the solution. – Clayton Feb 3 '13 at 14:12 A yes, thank you very much! – Francesca Feb 3 '13 at 14:13 If you like and understand the answer, feel free to accept it. If not, keep asking questions until you understand it clearly – Clayton Feb 3 '13 at 14:15 We're assuming$\;\log\;$denotes$\;\log = \ln\;$, i.e.,$\log$base$e$, where$\exp(x)$and$\log_e(x)$are inverse functions. Then remember that $$\exp(\log(a))=a\;\;\text{and}\;\;\exp(b+c)=\exp(a)\exp(c).$$ So$\exp( \log a + \log b) = \exp(\log a)\exp(\log b)= a\cdot b\$
Like it+ – Babak S. Feb 3 '13 at 15:24