What is 2+3log_{a}(a+1)-1/4 log_{a}(a-1) ?

The solution to 2+3log_{a}(a+1)-1/4 log_{a}(a-1) is 2+log_{a}(((a+1)^3(a-1)^{3/4})/(a-1))

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simplify 2+3log_{a}(a+1)-1/4 log_{a}(a-1)

simplify

2 + 3 lo g_{a} (a + 1) - \frac{1}{4} lo g_{a} (a - 1)

2 + lo g_{a} (\frac{( a + 1 ) ^{3} ( a - 1 ) ^{\frac{3}{4}}}{a - 1})

Solution steps

2 + 3 lo g_{a} (a + 1) - \frac{1}{4} lo g_{a} (a - 1)

Apply log rule:

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

3 lo g_{a} (a + 1) = lo g_{a} ((a + 1)^{3})

= 2 + lo g_{a} ((a + 1)^{3}) - \frac{1}{4} lo g_{a} (a - 1)

Apply log rule:

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

\frac{1}{4} lo g_{a} (a - 1) = lo g_{a} ((a - 1)^{\frac{1}{4}})

= 2 + lo g_{a} ((a + 1)^{3}) - lo g_{a} ((a - 1)^{\frac{1}{4}})

Apply log rule:

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

lo g_{a} ((a + 1)^{3}) - lo g_{a} ((a - 1)^{\frac{1}{4}}) = lo g_{a} (\frac{( a + 1 ) ^{3}}{( a - 1 ) ^{\frac{1}{4}}})

= 2 + lo g_{a} (\frac{( a + 1 ) ^{3}}{( a - 1 ) ^{\frac{1}{4}}})

Simplify

\frac{( a + 1 ) ^{3}}{( a - 1 ) ^{\frac{1}{4}}} : \frac{( a + 1 ) ^{3} ( a - 1 ) ^{\frac{3}{4}}}{a - 1}

= 2 + lo g_{a} (\frac{( a + 1 ) ^{3} ( a - 1 ) ^{\frac{3}{4}}}{a - 1})

lo g_{3} (20 x) - lo g_{6} (8 x^{3})

\frac{2 + ln ( 3 ) - ln ( 2 )}{2}

\frac{3 ln ( 2 )}{ln ( 3 )} - 2

ln (\frac{2}{x + 3})

4 lo g_{3} (u) - 20 lo g_{3} (v)

What is 2+3log_{a}(a+1)-1/4 log_{a}(a-1) ?
The solution to 2+3log_{a}(a+1)-1/4 log_{a}(a-1) is 2+log_{a}(((a+1)^3(a-1)^{3/4})/(a-1))