Help with a logarithm proof

Question

If $$\log \frac {1} {2} (a + b) = \frac {1} {2}(\log (a) + \log (b) )$$ Prove that $$ (a + b)^2 = 4ab $$

Can anyone show me how to do this?

Thanks.

score 3 · Accepted Answer · edited Mar 21 '18 at 21:02

3

$$2 \ln \left(\frac{1}{2} (a + b)\right) = \ln a + \ln b$$

$$\ln \left(\frac{1}{4} (a+b)^2\right) = \ln (ab)$$

$$(a+b)^2 = 4ab$$

edited Mar 21 '18 at 21:02

DonAntonio

answered Mar 21 '18 at 20:56

D F

1

I'll just add that you can go from the second line to the third because $ln(x)$ is an injective function. – F.A. Mar 21 '18 at 21:00
@F.A. yes, a useful remark – D F Mar 21 '18 at 21:02
How did you go from $ \frac {1} {2} (a+b) $ to $ \frac {1} {4} (a+b)^2 $ ? – Jimmy Mar 21 '18 at 21:09
@Jimmy because of the identity $a \ln b = \ln (b^a)$ – D F Mar 21 '18 at 21:10
Okay I see. Thank you. – Jimmy Mar 21 '18 at 21:12
@Jimmy you're welcome! – D F Mar 21 '18 at 21:14

1 Answers1