(log(x^6)+2log(y^3))/log(xy)
remember a*log(b)= log(b^a)
and log a+ log b= log (ab)
and log a/logb= log (a)-log (b)
recall that when the expression inside the log is raised to a certain power, we can write the exponent outside of the log, for instance:
log (x^2) = 2 log x
log [(2y)^0.5] = 0.5 log 2y
thus for the problem, we can rewrite it as:
(log(x^6) + 2log(y^3)) / log(xy)
(6 log x + 6 log y ) / log (xy)
6 (log x + log y) / log (xy)
now recall that the sum of log with the same base is the same as the log of the product of the expressions inside the parenthesis, for instance:
log 2 + log x = log 2x
log a + log b = log ab
thus we can further simplify this as:
6 (log x + log y) / log (xy)
6 (log xy) / log xy
cancelling out the log terms, we have 6
hope this helps~ :)
To simplify the given expression (log(x^6) + 2log(y^3))/(log(xy)), we can start by applying the properties of logarithms.
First, let's use the power property of logarithms, which states that log(a^b) = b * log(a):
log(x^6) = 6 * log(x)
log(y^3) = 3 * log(y)
By substituting these values back into the original expression, we get:
(6 * log(x) + 2 * 3 * log(y)) / log(xy)
Next, let's simplify the expression further:
(6 * log(x) + 6 * log(y)) / log(xy)
Now, we can use the distributive property to factor out a common term of 6:
(6 * (log(x) + log(y))) / log(xy)
Since log(x) + log(y) is equal to log(xy) according to the product property of logarithms, we can simplify further:
(6 * log(xy)) / log(xy)
Finally, log(xy) appears in the numerator and denominator, so they cancel out:
6
Therefore, the simplified form of the expression (log(x^6) + 2log(y^3))/(log(xy)) is 6.