What is 10ln(x^2y^5z^3)-λ(5x+2y+8z) ?

The solution to 10ln(x^2y^5z^3)-λ(5x+2y+8z) is 20ln(x)+50ln(y)+30ln(z)-λ(5x+2y+8z)

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simplify 10ln(x^2y^5z^3)-λ(5x+2y+8z)

simplify

10 ln (x^{2} y^{5} z^{3}) - λ (5 x + 2 y + 8 z)

20 ln (x) + 50 ln (y) + 30 ln (z) - λ (5 x + 2 y + 8 z)

Solution steps

10 ln (x^{2} y^{5} z^{3}) - λ (5 x + 2 y + 8 z)

Apply log rule:

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

10 ln (x^{2} y^{5} z^{3}) = 10 ln (x^{2}) + 10 ln (y^{5}) + 10 ln (z^{3})

= 10 ln (x^{2}) + 10 ln (y^{5}) + 10 ln (z^{3}) - λ (5 x + 2 y + 8 z)

Apply log rule:

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

ln (x^{2}) = 2 ln (x)

= 10 \cdot 2 ln (x) + 10 ln (y^{5}) + 10 ln (z^{3}) - λ (5 x + 2 y + 8 z)

Apply log rule:

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

ln (y^{5}) = 5 ln (y)

= 10 \cdot 2 ln (x) + 10 \cdot 5 ln (y) + 10 ln (z^{3}) - λ (5 x + 2 y + 8 z)

Apply log rule:

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

ln (z^{3}) = 3 ln (z)

= 10 \cdot 2 ln (x) + 10 \cdot 5 ln (y) + 10 \cdot 3 ln (z) - λ (5 x + 2 y + 8 z)

Multiply the numbers:

10 \cdot 2 = 20

= 20 ln (x) + 10 \cdot 5 ln (y) + 10 \cdot 3 ln (z) - λ (5 x + 2 y + 8 z)

Multiply the numbers:

10 \cdot 5 = 50

= 20 ln (x) + 50 ln (y) + 10 \cdot 3 ln (z) - λ (5 x + 2 y + 8 z)

Multiply the numbers:

10 \cdot 3 = 30

= 20 ln (x) + 50 ln (y) + 30 ln (z) - λ (5 x + 2 y + 8 z)

lo g_{10} (y x^{5})

7 lo g_{3} (u) + 4 lo g_{3} (v)

- 1.46 - 5 lo g_{10} (\frac{152.67}{10})

3 ln (x) - 5 ln (\frac{1}{y}) - 4 ln (x y)

lo g_{8} (x \cdot y \cdot z^{6})

What is 10ln(x^2y^5z^3)-λ(5x+2y+8z) ?
The solution to 10ln(x^2y^5z^3)-λ(5x+2y+8z) is 20ln(x)+50ln(y)+30ln(z)-λ(5x+2y+8z)