Question

express log (base b) 2 +log (base b) beta + (1/2)log (base b)Y- (1/2)log(base b) w as a single logarithm?

(P.S: I think that w is an angular speed symbol)
Thank you!! sooo much for reading and repling!!

Answer 1

Remember, the sum of logarithms with the same base is the same as the logarithm of all those things inside multiplied together. That is:\[\log_{a}(x)+\log_{a}(y)+\log_{a}(z)...\log_{a}(n)=\log_{a}(x \times y \times z ...\times n) \]We also know that if the logarithms were being subtracted instead of being added, then instead of multiplying, we would divide them. And we also know that the exponent of a logarithm can be brought over to the front and be multiplied by the whole thing.

From this we can re-write your given logarithm as:
\[\log_{b}(2) +\log_{b}(\beta) +\frac{ 1 }{ 2 }\log_{b}(Y)-\frac{ 1 }{ 2 }\log_{b}w=\log_{b}(2 \times \beta \times Y^{1/2} )-\log_{b} w^{1/2}\]\[=\log_{b}\left( \frac{ 2 \times \beta \times Y^{1/2} }{ w^{1/2} } \right) =\log_{b}\left( \frac{ 2 \beta \sqrt{ Y} }{ \sqrt{w} } \right)\]
@Tayseer

Answer 2

Yes! I understand now! Thank You sooo much! You are the best