What is log_{1/3}(1/3 (x^2+2x)) ?

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

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simplify log_{1/3}(1/3 (x^2+2x))

simplify

lo g_{\frac{1}{3}} (\frac{1}{3} (x^{2} + 2 x))

- lo g_{3} (x^{2} + 2 x) + 1

Solution steps

lo g_{\frac{1}{3}} (\frac{1}{3} (x^{2} + 2 x))

Apply log rule:

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

lo g_{\frac{1}{3}} (\frac{1}{3} (x^{2} + 2 x)) = lo g_{\frac{1}{3}} (\frac{1}{3}) + lo g_{\frac{1}{3}} (x^{2} + 2 x)

= lo g_{\frac{1}{3}} (\frac{1}{3}) + lo g_{\frac{1}{3}} (x^{2} + 2 x)

Apply log rule:

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

lo g_{\frac{1}{3}} (\frac{1}{3}) = lo g_{\frac{1}{3}} (1) - lo g_{\frac{1}{3}} (3)

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

Apply log rule:

lo g_{a} (1) = 0

= 0 - lo g_{\frac{1}{3}} (3) + lo g_{\frac{1}{3}} (x^{2} + 2 x)

0 - lo g_{\frac{1}{3}} (3) + lo g_{\frac{1}{3}} (x^{2} + 2 x) = - lo g_{\frac{1}{3}} (3) + lo g_{\frac{1}{3}} (x^{2} + 2 x)

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

Apply log rule:

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

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

lo g_{\frac{1}{3}} (3) = - 1

= - lo g_{3} (x^{2} + 2 x) - (- 1)

Apply rule

- (- a) = a

= - lo g_{3} (x^{2} + 2 x) + 1

lo g_{3} (\frac{4 x ^{2}}{y})

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

ln (x^{2} e^{b x})

lo g_{2} (\frac{1}{4 x ^{2}})

lo g_{3} (9 x^{3})

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