What is ln((3xsqrt(1+5x))/((x-6)^8)) ?

The solution to ln((3xsqrt(1+5x))/((x-6)^8)) is ln(3)+ln(x)+ln(sqrt(1+5x))-8ln(x-6)

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simplify ln((3xsqrt(1+5x))/((x-6)^8))

simplify

ln (\frac{3 x 1 + 5 x}{( x - 6 ) ^{8}})

ln (3) + ln (x) + ln (1 + 5 x) - 8 ln (x - 6)

Solution steps

ln (\frac{3 x 1 + 5 x}{( x - 6 ) ^{8}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

ln (\frac{3 x 1 + 5 x}{( x - 6 ) ^{8}}) = ln (3 x 1 + 5 x) - ln ((x - 6)^{8})

= ln (3 x 5 x + 1) - ln ((x - 6)^{8})

Apply log rule:

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

ln ((x - 6)^{8}) = 8 ln (x - 6)

= ln (3 x 1 + 5 x) - 8 ln (x - 6)

Apply log rule:

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

ln (3 x 1 + 5 x) = ln (3) + ln (x) + ln (1 + 5 x)

= ln (3) + ln (x) + ln (5 x + 1) - 8 ln (x - 6)

ln (\frac{- 1}{2})

lo g_{7} (x y^{3})

- lo g_{10} (4 \times 1 0^{- 3})

ln (3 x)

ln (x (x + 4) (x + 5))

What is ln((3xsqrt(1+5x))/((x-6)^8)) ?
The solution to ln((3xsqrt(1+5x))/((x-6)^8)) is ln(3)+ln(x)+ln(sqrt(1+5x))-8ln(x-6)