\log \left(\frac{75}{16}\right)-2 \log \left(\frac{5}{9}\right)+\log 3

1.	\log \left(\frac{75}{16}\right)-2 \log \left(\frac{5}{9}\right)+\log 32-\log 243-\log 2
Answer» Given log(75/16) - 2log(5/9) + log(32/243). ------ (1) We know that a log(b) = log b^a. 2 log(5/9) = log (5/9)^2. We know that log(a) - log(b) = log(a/b). log(75/16) - log(5/9)^2 = log(75/16/(5/9)^2. = log (75/400/81) = log (75 * 31/400) = log (6075/400) = log (243/16) ----- (2) Substitute (2) in (1), we get = log (243/16) + log(32/243) We know that log a + log b = log ab. log(243/16) + log(32/243) = log(243/16 * log 32/243) = log(243 * 32/16 * 243) = log(32/16) = log 2. but answer is 0

\log \left(\frac{75}{16}\right)-2 \log \left(\frac{5}{9}\right)+\log 32-\log 243-\log 2

Answer»

Given log(75/16) - 2log(5/9) + log(32/243). ------ (1)

We know that a log(b) = log b^a.

2 log(5/9) = log (5/9)^2.

We know that log(a) - log(b) = log(a/b).

log(75/16) - log(5/9)^2 = log(75/16/(5/9)^2.

= log (75/400/81)

= log (75 * 31/400)

= log (6075/400)

= log (243/16) ----- (2)

Substitute (2) in (1), we get

= log (243/16) + log(32/243)

We know that log a + log b = log ab.

log(243/16) + log(32/243) = log(243/16 * log 32/243)

= log(243 * 32/16 * 243)

= log(32/16)

= log 2.

but answer is 0