What is ln((8^x(3x^3+6)^2)^8(9x^4-1)^2) ?

The solution to ln((8^x(3x^3+6)^2)^8(9x^4-1)^2) is 24ln(2)x+16ln(3x^3+6)+2ln(9x^4-1)

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simplify ln((8^x(3x^3+6)^2)^8(9x^4-1)^2)

simplify

ln ((8^{x} (3 x^{3} + 6)^{2})^{8} (9 x^{4} - 1)^{2})

24 ln (2) x + 16 ln (3 x^{3} + 6) + 2 ln (9 x^{4} - 1)

Solution steps

ln ((8^{x} (3 x^{3} + 6)^{2})^{8} (9 x^{4} - 1)^{2})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

ln ((8^{x} (3 x^{3} + 6)^{2})^{8} (9 x^{4} - 1)^{2}) = ln ((8^{x} (3 x^{3} + 6)^{2})^{8}) + ln ((9 x^{4} - 1)^{2})

= ln ((8^{x} (3 x^{3} + 6)^{2})^{8}) + ln ((9 x^{4} - 1)^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((8^{x} (3 x^{3} + 6)^{2})^{8}) = 8 ln (8^{x} (3 x^{3} + 6)^{2})

= 8 ln (8^{x} (3 x^{3} + 6)^{2}) + ln ((9 x^{4} - 1)^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((9 x^{4} - 1)^{2}) = 2 ln (9 x^{4} - 1)

= 8 ln (8^{x} (3 x^{3} + 6)^{2}) + 2 ln (9 x^{4} - 1)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

8 ln (8^{x} (3 x^{3} + 6)^{2}) = 8 ln (8^{x}) + 8 ln ((3 x^{3} + 6)^{2})

= 8 ln (8^{x}) + 8 ln ((3 x^{3} + 6)^{2}) + 2 ln (9 x^{4} - 1)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (8^{x}) = x ln (8)

= 8 x ln (8) + 8 ln ((3 x^{3} + 6)^{2}) + 2 ln (9 x^{4} - 1)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln ((3 x^{3} + 6)^{2}) = 2 ln (3 x^{3} + 6)

= 8 x ln (8) + 8 \cdot 2 ln (3 x^{3} + 6) + 2 ln (9 x^{4} - 1)

Multiply the numbers:

8 \cdot 2 = 16

= 8 ln (8) x + 16 ln (3 x^{3} + 6) + 2 ln (9 x^{4} - 1)

8 x ln (8) = 24 ln (2) x

= 24 ln (2) x + 16 ln (3 x^{3} + 6) + 2 ln (9 x^{4} - 1)

10 ln (\frac{50}{1 0 ^{- 12}})

5.23 + lo g_{10} (\frac{1.76 \cdot 1 0 ^{- 3}}{0.00391})

7 ln (9 x)

lo g_{4} (\frac{3 x}{y})

ln (2) + ln (x - 2)

What is ln((8^x(3x^3+6)^2)^8(9x^4-1)^2) ?
The solution to ln((8^x(3x^3+6)^2)^8(9x^4-1)^2) is 24ln(2)x+16ln(3x^3+6)+2ln(9x^4-1)