What is 1/2 [6ln(x-3)+ln(x)-2ln(x)] ?

The solution to 1/2 [6ln(x-3)+ln(x)-2ln(x)] is 1/2 ln(((x-3)^6)/x)

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simplify 1/2 [6ln(x-3)+ln(x)-2ln(x)]

simplify

\frac{1}{2} [6 ln (x - 3) + ln (x) - 2 ln (x)]

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

Solution steps

\frac{1}{2} [6 ln (x - 3) + ln (x) - 2 ln (x)]

Apply log rule:

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

6 ln (x - 3) = ln ((x - 3)^{6})

= \frac{1}{2} [ln ((x - 3)^{6}) + ln (x) - 2 ln (x)]

Apply log rule:

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

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

= \frac{1}{2} [ln ((x - 3)^{6} x) - 2 ln (x)]

Apply log rule:

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

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

= \frac{1}{2} [ln (x (x - 3)^{6}) - ln (x^{2})]

Apply log rule:

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

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

= \frac{1}{2} [ln (\frac{x ( x - 3 ) ^{6}}{x ^{2}})]

Cancel the common factor:

x

= \frac{1}{2} [ln (\frac{( x - 3 ) ^{6}}{x})]

Remove parentheses:

(a) = a

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

lo g_{10} (6) + lo g_{10} (3)

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

- lo g_{10} (3.8 \cdot 1 0^{- 4})

2 lo g_{10} (x) - 3 lo g_{10} (x y) - 4 lo g_{10} (y)

lo g_{4} (x^{2} - 64) - 6 lo g_{4} (x + 8)

What is 1/2 [6ln(x-3)+ln(x)-2ln(x)] ?
The solution to 1/2 [6ln(x-3)+ln(x)-2ln(x)] is 1/2 ln(((x-3)^6)/x)