What is log_{a}((3^4)/(4^2a^7)) ?

The solution to log_{a}((3^4)/(4^2a^7)) is 4log_{a}(3)-4log_{a}(2)-7

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simplify log_{a}((3^4)/(4^2a^7))

simplify

lo g_{a} (\frac{3 ^{4}}{4 ^{2} a ^{7}})

4 lo g_{a} (3) - 4 lo g_{a} (2) - 7

Solution steps

lo g_{a} (\frac{3 ^{4}}{4 ^{2} a ^{7}})

Simplify

\frac{3 ^{4}}{4 ^{2} a ^{7}} : \frac{81}{16 a ^{7}}

= lo g_{a} (\frac{81}{16 a ^{7}})

Apply log rule:

lo g_{a} (\frac{x}{y}) = lo g_{a} (x) - lo g_{a} (y)

lo g_{a} (\frac{81}{16 a ^{7}}) = lo g_{a} (81) - lo g_{a} (16 a^{7})

= lo g_{a} (81) - lo g_{a} (16 a^{7})

lo g_{a} (81) = 4 lo g_{a} (3)

lo g_{a} (16 a^{7}) = 4 lo g_{a} (2) + 7

= 4 lo g_{a} (3) - (4 lo g_{a} (2) + 7)

Apply the distributive law:

- (a + b) = - a - b

- (4 lo g_{a} (2) + 7) = - 4 lo g_{a} (2) - 7

= 4 lo g_{a} (3) - 4 lo g_{a} (2) - 7

ln (x^{\frac{1}{2}} y^{9}) - ln (z^{4})

ln (x) + ln (y) - 5 ln (z)

ln ((x + 3)^{3}) - 2 ln (x) - 2 ln (x^{2} + 3 x)

lo g_{10} (7 x^{4})

7 lo g_{a} (m) - \frac{1}{3} lo g_{a} (r)

What is log_{a}((3^4)/(4^2a^7)) ?
The solution to log_{a}((3^4)/(4^2a^7)) is 4log_{a}(3)-4log_{a}(2)-7