Show that logx+log(xy)+log(xy^2)is the arthematic progression
log ( x ∙ y ) = log x + log y
log ( x ∙ y² ) = log x + log ( y² ) = log x + 2 log y
log x + log ( x ∙ y ) + log ( x ∙ y² ) = log x + log x + log y + log x + 2 log y
In A.P:
an = a1 + ( n - 1 ) ∙ d
in this case:
a1 = log x , d = log y
Proof:
a1 = log x
a2 = a1 + ( 2 - 1 ) ∙ d = a1 + 1 ∙ d = a1 + d = log x + log y
a3 = a1 + ( 3 - 1 ) ∙ d = a1 + 2 ∙ d = a1 + 2 d = log x + 2 log y
So log x + log ( x ∙ y ) + log ( x ∙ y² ) is arithmetic progression.
If a, b, and c are in an arithmetic progression, then b-a = c - b
so we have to show that log(xy) - logx = log (xy^2) - log (xy)
LS = log xy - logx
= logx + logy - logx = logy
RS = log(xy^2) - log(xy)
= logx + 2logy - logx - logy = logy
LS = RS
Yes, it is an AP
To show that the given expression is an arithmetic progression, we need to find the common difference between consecutive terms.
We have the expression:
log(x) + log(xy) + log(xy^2)
Using the properties of logarithms, we can simplify this expression:
log(x) + log(xy) + log(xy^2) = log(x) + log(x) + log(y) + log(xy) + log(y) + log(y)
Next, we combine the logarithms using the logarithmic identity log(a) + log(b) = log(ab):
= log(x * x * y * xy * y * y)
= log(x^2 * y^4)
Now, we can rewrite the expression in terms of exponents to see if it is an arithmetic progression:
log(x^2 * y^4) = 2log(x) + 4log(y)
We can see that the logarithms of x and y have coefficients of 2 and 4, respectively. In an arithmetic progression, the common difference is the same between terms. In this case, the common difference is 2 since the coefficient of log(x) changes by 2 and the coefficient of log(y) changes by 4.
Hence, the expression log(x) + log(xy) + log(xy^2) is an arithmetic progression with a common difference of 2.
To show that log x + log(xy) + log(xy^2) is an arithmetic progression, we need to determine if the difference between consecutive terms is constant.
The given expression can be simplified using logarithm properties: log(x) + log(xy) + log(xy^2) = log(x) + log(xy^3).
Next, we can use the logarithmic identity log(a) + log(b) = log(ab). Applying this identity, we have: log(x) + log(xy^3) = log(x * xy^3) = log(x^2y^3).
So, the given expression simplifies to log(x^2y^3).
Now, let's analyze the terms in the expression:
- The first term, log(x), has a coefficient of 1.
- The second term, log(xy^3), has a coefficient of 2 (x^2y^3 = (xy^3)^2).
- The third term, log(x^2y^3), has a coefficient of 3.
Notice that the coefficients are consecutive numbers: 1, 2, 3. This demonstrates that log x + log(xy) + log(xy^2) forms an arithmetic progression.
In conclusion, the given expression log x + log(xy) + log(xy^2) can be represented as the arithmetic progression log(x), log(xy^3), log(x^2y^3) with a difference of 1 between consecutive terms.