Express in single logarithm
2loga x - 3loga y - 4
loga (x^2/y^3) - 4
= loga (x^2/(a^4y^3))
(using the property loga (b/c) = loga b - loga c)
= loga (x^2/a^4) - loga y^3 - 4
(using the property loga (b*c) = loga b + loga c)
= loga (x^2/a^4) + loga (1/y^3) - 4
(using the property loga 1 = 0)
= loga (x^2/a^4) + loga (1/y^3) - loga a^4
(using the property loga b - loga c = loga (b/c))
= loga [(x^2/a^4) * (1/y^3) * (1/a^4)]
To express the expression 2loga(x) - 3loga(y) - 4 in a single logarithm, we can use logarithmic identities.
Start by rewriting the expression using the properties of logarithms:
2loga(x) - 3loga(y) - 4
Now, let's simplify this expression step by step:
Step 1: Apply the power rule of logarithms
loga(x^2) - loga(y^3) - 4
Step 2: Apply the quotient rule of logarithms to combine the logarithms with the same base.
loga((x^2)/(y^3)) - 4
Therefore, the expression 2loga(x) - 3loga(y) - 4 can be simplified to loga((x^2)/(y^3)) - 4.