Help with equations, logs and ln

Given y = 4^(x^2 - 5x) * 6^(x^3 - 4x^2) * 7^(5x+4), ln(y) can be written ln(y) = ln(A)x^3 + ln(B)x^2 + ln(C)x +ln(D). What are the values of A, B, C and D?

I've been trying this one for a while now, and I'm stumped! Any help would be appreciated! I don't need an answer, but some steps would help greatly. Thanks!

So first step is obviously to take ln of both sides of the original equation.

ln(y)=ln(4^(x^2 - 5x) * 6^(x^3 - 4x^2) * 7^(5x+4))

Now you can split up the parts that are multiplied together because ln(A*B)=ln(A)+ln(B)

ln(4^(x^2 - 5x))+ln(6^(x^3 - 4x^2))+ln(7^(5x+4))

Now bring down the exponential part because ln(A^B)=B*ln(A)

(x^2 - 5x)ln(4)+(x^3 - 4x^2)ln(6)+(5x+4)ln(7)

Now expand everything.

x^2ln(4)-5xln(4)+x^3ln(6)-4x^2ln(6)+5x…

Not factor out the different x powers.

x^3ln(6)+x^2(ln(4)-4ln(6))+x(-5ln(4)+5…

Now use the same ln rules from earlier but in reverse to put everything back together.

x^3ln(6)+x^2ln(4*6^-4)+xln(4^-5*7^5)+l…

x^3ln(6)+x^2ln(1/324)+xln(16807/1024)+…

A=6 B=1/324 C=16807/1024 D=2401

ln(y) = ln(4^(x² - 5x) * 6^(x³ - 4x²) * 7^(5x+4))
ln(y) = ln(4^(x² - 5x)) + ln(6^(x³ - 4x²)) + ln(7^(5x+4))
ln(y) = (x² - 5x) ln(4) + (x³ - 4x²) ln(6) + (5x+4) ln(7)
ln(y) = x² ln(4) - 5x ln(4) + x³ ln(6) - 4x² ln(6) + 5x ln(7) + 4 ln(7)
ln(y) = x³ ln(6) + x² (ln(4) - 4 ln(6)) + x (5 ln(7) - 5 ln(4)) + 4 ln(7)
ln(y) = x³ ln(6) + x² (ln(4) - ln(6⁴)) + x (ln(7⁵) - ln(4⁵)) + ln(7⁴)
ln(y) = ln(6) x³ + ln(4/6⁴) x² + ln(7⁵/4⁵) x + ln(7⁴)
ln(y) = ln(6) x³ + ln(1/324) x² + ln(16807/1024) x + ln(2401)

A = 6
B = 1/324
C = 16807/1024
D = 2401

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NOTE: Your previous question was deleted for some reason:

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