What is 1/3 ln((x-4)^6)+ln(x)-ln(x^2-8x+16) ?

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

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simplify 1/3 ln((x-4)^6)+ln(x)-ln(x^2-8x+16)

simplify

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

ln (x)

Solution steps

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln ((x - 4)^{6}) = ln (((x - 4)^{6})^{\frac{1}{3}})

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

Apply log rule:

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

ln (((x - 4)^{6})^{\frac{1}{3}}) + ln (x) = ln (x ((x - 4)^{6})^{\frac{1}{3}})

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

((x - 4)^{6})^{\frac{1}{3}} = (x - 4)^{2}

= ln (x (x - 4)^{2}) - ln (x^{2} - 8 x + 16)

Apply log rule:

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

ln (x (x - 4)^{2}) - ln (x^{2} - 8 x + 16) = ln (\frac{x ( x - 4 ) ^{2}}{x ^{2} - 8 x + 16})

= ln (\frac{x ( x - 4 ) ^{2}}{x ^{2} - 8 x + 16})

Simplify

\frac{x ( x - 4 ) ^{2}}{x ^{2} - 8 x + 16} : x

= ln (x)

6 lo g_{b} (t) - lo g_{b} (s)

ln (x y)

lo g_{4} (9) - lo g_{4} (y)

- lo g_{10} (6.9 \cdot 1 0^{- 8})

lo g_{3} (m) + lo g_{3} (m + 1) - lo g_{2} (m + 6)

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