What is 7log_{3}(a)+2log_{3}(b)-1/2 log_{3}(2c) ?

The solution to 7log_{3}(a)+2log_{3}(b)-1/2 log_{3}(2c) is log_{3}((a^7b^2)/(sqrt(2)\sqrt{c)})

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simplify 7log_{3}(a)+2log_{3}(b)-1/2 log_{3}(2c)

simplify

7 lo g_{3} (a) + 2 lo g_{3} (b) - \frac{1}{2} lo g_{3} (2 c)

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

Solution steps

7 lo g_{3} (a) + 2 lo g_{3} (b) - \frac{1}{2} lo g_{3} (2 c)

7 lo g_{3} (a) = lo g_{3} (a^{7})

2 lo g_{3} (b) = lo g_{3} (b^{2})

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

= lo g_{3} (a^{7}) + lo g_{3} (b^{2}) - lo g_{3} ((2 c)^{\frac{1}{2}})

Apply log rule:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

lo g_{3} (a^{7}) + lo g_{3} (b^{2}) = lo g_{3} (a^{7} b^{2})

= lo g_{3} (a^{7} b^{2}) - lo g_{3} ((2 c)^{\frac{1}{2}})

Apply log rule:

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

lo g_{3} (a^{7} b^{2}) - lo g_{3} ((2 c)^{\frac{1}{2}}) = lo g_{3} (\frac{a ^{7} b ^{2}}{( 2 c ) ^{\frac{1}{2}}})

= lo g_{3} (\frac{a ^{7} b ^{2}}{( 2 c ) ^{\frac{1}{2}}})

Simplify

\frac{a ^{7} b ^{2}}{( 2 c ) ^{\frac{1}{2}}} : \frac{a ^{7} b ^{2}}{2 ^{\frac{1}{2}} c ^{\frac{1}{2}}}

= lo g_{3} (\frac{a ^{7} b ^{2}}{2 ^{\frac{1}{2}} c ^{\frac{1}{2}}})

Apply exponent rule:

a^{\frac{1}{2}} = a

2^{\frac{1}{2}} = 2

= lo g_{3} (\frac{a ^{7} b ^{2}}{2 c})

Apply exponent rule:

a^{\frac{1}{2}} = a

c^{\frac{1}{2}} = c

= lo g_{3} (\frac{a ^{7} b ^{2}}{2 c})

lo g_{10} (a) - 2 lo g_{10} (b) + 3 lo g_{10} (c)

lo g_{10} (a) + 2 - 2 lo g_{10} (b)

4 lo g_{3} (x) + 7 lo g_{3} (y)

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

ln (\frac{x}{x ^{6} y ^{6}})

What is 7log_{3}(a)+2log_{3}(b)-1/2 log_{3}(2c) ?
The solution to 7log_{3}(a)+2log_{3}(b)-1/2 log_{3}(2c) is log_{3}((a^7b^2)/(sqrt(2)\sqrt{c)})