What is 5ln(x-2)+3ln(4)-ln(8)-2ln(x)-1/2 ln(81) ?

The solution to 5ln(x-2)+3ln(4)-ln(8)-2ln(x)-1/2 ln(81) is ln((8(x-2)^5)/(9x^2))

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simplify 5ln(x-2)+3ln(4)-ln(8)-2ln(x)-1/2 ln(81)

simplify

5 ln (x - 2) + 3 ln (4) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

ln (\frac{8 ( x - 2 ) ^{5}}{9 x ^{2}})

Solution steps

5 ln (x - 2) + 3 ln (4) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

Apply log rule:

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

5 ln (x - 2) = ln ((x - 2)^{5})

= ln ((x - 2)^{5}) + 3 ln (4) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

Apply log rule:

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

3 ln (4) = ln (4^{3})

= ln ((x - 2)^{5}) + ln (4^{3}) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

Apply log rule:

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

ln ((x - 2)^{5}) + ln (4^{3}) = ln (4^{3} (x - 2)^{5})

= ln (4^{3} (x - 2)^{5}) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

4^{3} = 64

= ln (64 (x - 2)^{5}) - ln (8) - 2 ln (x) - \frac{1}{2} ln (81)

Apply log rule:

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

ln (64 (x - 2)^{5}) - ln (8) = ln (\frac{64 ( x - 2 ) ^{5}}{8})

= ln (\frac{64 ( x - 2 ) ^{5}}{8}) - 2 ln (x) - \frac{1}{2} ln (81)

Divide the numbers:

\frac{64}{8} = 8

= ln (8 (x - 2)^{5}) - 2 ln (x) - \frac{1}{2} ln (81)

Apply log rule:

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

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

= ln (8 (x - 2)^{5}) - ln (x^{2}) - \frac{1}{2} ln (81)

Apply log rule:

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

ln (8 (x - 2)^{5}) - ln (x^{2}) = ln (\frac{8 ( x - 2 ) ^{5}}{x ^{2}})

= ln (\frac{8 ( x - 2 ) ^{5}}{x ^{2}}) - \frac{1}{2} ln (81)

Apply log rule:

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

\frac{1}{2} ln (81) = ln (8 1^{\frac{1}{2}})

= ln (\frac{8 ( x - 2 ) ^{5}}{x ^{2}}) - ln (8 1^{\frac{1}{2}})

Apply log rule:

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

ln (\frac{8 ( x - 2 ) ^{5}}{x ^{2}}) - ln (8 1^{\frac{1}{2}}) = ln (\frac{\frac{8 ( x - 2 ) ^{5}}{x ^{2}}}{8 1 ^{\frac{1}{2}}})

= ln (\frac{\frac{8 ( x - 2 ) ^{5}}{x ^{2}}}{8 1 ^{\frac{1}{2}}})

Simplify

\frac{\frac{8 ( x - 2 ) ^{5}}{x ^{2}}}{8 1 ^{\frac{1}{2}}} : \frac{8 ( x - 2 ) ^{5}}{9 x ^{2}}

= ln (\frac{8 ( x - 2 ) ^{5}}{9 x ^{2}})

\frac{\frac{423324672 \cdot 6}{( 3 - 3 )}}{ln ( 1 + 3 ) + ln ( 1 - 3 )}

- lo g_{10} (7.3 X 1 0^{- 5})

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

4 lo g_{10} (x) + 4 lo g_{10} (y)

4 lo g_{10} (x) - 6 lo g_{10} (y)

What is 5ln(x-2)+3ln(4)-ln(8)-2ln(x)-1/2 ln(81) ?
The solution to 5ln(x-2)+3ln(4)-ln(8)-2ln(x)-1/2 ln(81) is ln((8(x-2)^5)/(9x^2))