+9519 \(\text{Given that }a=\log\left(1+\frac{1}{3}\right)\space\text{and}\space b=\log\left(1+\frac{1}{9}\right), \\\text{express log 2 and log 3 in terms of a and b.}\)
+33616 Can also write
a = log(4/3) → log4 - log3 → log(2^2) - log3 → 2log2 - log3
and
b = log(10/9) → log10 - log9 → 1 - log(3^2) → 1 - 2log3. (I'm assuming log is log to base 10)
From these you should be able to rearrange to get log2 and log3 in terms of a and b.
+128475 For the first one, we have
a = log ( 1 + 1/3) → log (4/3) → log (4) - log (3) → log(2^2) - log (3) = 2log 2 - log (3)
Add log(3) to both sides
a + log(3) = 2log(2)
Divide both sides by 2
[ a + log (3) ] / 2 = log (2)
For the second we have
b = log (1 + 1/9) → log(10/9) → log (10) - log (9) → 1 - log(3^2) → 1 - 2log(3)
Add 2log(3) to both sides and subtract b from both sides
2log(3) = 1 - b divide both sides by 2
log (3) = [ 1 - b ] / 2
