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How do you condense #1/3log 3x + 2/3log 3x #?

Answer 1

Sadie Alexander

#1/3 log 3x + 2/3 log 3x##=log(3x)#

#1/3 log 3x + 2/3 log 3x#

#= log(3x)^(1/3) + log(3x)^(2/3)#-> use property #log_bx ^n = nlog _b x#

#=log((3x)^(1/3) (3x)^(2/3))#-> use property #log_b(xy)=log_bx+log_by#

#=log((3x)^(1/3+2/3))#-> use property #x^a*x^b=x^(a+b)#

#=log(3x)#

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Answer 2

Ariana Alvarez

To condense the expression ( \frac{1}{3} \log(3x) + \frac{2}{3} \log(3x) ), you can combine the logarithms using the properties of logarithms, specifically the property that states:

[ \log_a(x) + \log_a(y) = \log_a(xy) ]

Applying this property, we have:

[ \frac{1}{3} \log(3x) + \frac{2}{3} \log(3x) = \log(3x)^{\frac{1}{3}} + \log(3x)^{\frac{2}{3}} ]

Now, since ( \log_a(x^m) = m \log_a(x) ), we can rewrite this as:

[ \log((3x)^{\frac{1}{3}} \cdot (3x)^{\frac{2}{3}}) ]

Simplify the expression inside the logarithm:

[ \log((3x)^{\frac{1}{3} + \frac{2}{3}}) ]

[ = \log((3x)^1) ]

[ = \log(3x) ]

So, the condensed form of the expression is ( \log(3x) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.