Use the rules of logarithms to expand [(x+3)^4(x-5)^7]
A poorly worded question,
we would not use logs to expand this
perhaps you meant:
if y = [(x+3)^4(x-5)^7]
then log y = log (x+3)^4 + log (x-5)^7
= 4log(x+3) + 7log(x-5)
btw, here are the 12 simplified terms of the expansion.
http://www.wolframalpha.com/input/?i=expand+%28x%2B3%29%5E4%28x-5%29%5E7
okay thank you :)
To expand the expression [(x+3)^4(x-5)^7] using the rules of logarithms, we can first use the power rule of exponents to expand each factor separately.
Let's start with (x+3)^4. According to the power rule, when a term is raised to a power, we can distribute that power to each term inside the parentheses. In this case, we have:
(x+3)^4 = x^4 + 4x^3 * 3 + 6x^2 * 3^2 + 4x * 3^3 + 3^4
Expanding this further:
(x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81
Now let's expand (x-5)^7 using the same rule:
(x-5)^7 = x^7 - 7x^6 * 5 + 21x^5 * 5^2 - 35x^4 * 5^3 + 35x^3 * 5^4 - 21x^2 * 5^5 + 7x * 5^6 - 5^7
Simplifying:
(x-5)^7 = x^7 - 35x^6 + 525x^5 - 4375x^4 + 21875x^3 - 65625x^2 + 109375x - 78125
To obtain the expanded form of [(x+3)^4(x-5)^7], we can multiply the two expanded expressions:
[(x+3)^4(x-5)^7] = (x^4 + 12x^3 + 54x^2 + 108x + 81) * (x^7 - 35x^6 + 525x^5 - 4375x^4 + 21875x^3 - 65625x^2 + 109375x - 78125)
We can further simplify this expression by using the distributive property to multiply each term in the first expression by each term in the second expression. This may result in a long and detailed equation, but it can be simplified and organized by combining like terms if desired.
Keep in mind that expanding and simplifying expressions can get quite complex, especially for higher powers or more terms.