Question

HELP PLEASEEEE
write the expression
3log(x-2)+log(x+2)-log(x^2-4) as single logarithm

Answer 1

Here, we can apply these properties of logarithms:
\[\huge \begin{gathered}
k\log \left( x \right) = \log \left( {{x^k}} \right) \hfill \\
\hfill \\
\log \left( x \right) + \log \left( y \right) = \log \left( {xy} \right) \hfill \\
\hfill \\
\log \left( x \right) - \log \left( y \right) = \log \left( {\frac{x}{y}} \right) \hfill \\
\end{gathered} \]
being \(k\) any real number, and \(x>0,\;y>0\)

Answer 2

Okay, yes I know that rules.. I think I know the answer but am not sure. Should I multiply 3log(x-2)+log(x+2)? Or should I turn 3log(x-2) into log(x-2^3) first?

Answer 3

@Michele_Laino

Answer 4

come back please

Answer 5

first step:
I apply this property:
\[\huge k\log \left( x \right) = \log \left( {{x^k}} \right)\]
so, I can write:
\[\huge 3\log \left( {x - 2} \right) = \log \left\{ {{{\left( {x - 2} \right)}^3}} \right\}\]

Answer 6

okay that's what i did too :)

Answer 7

now, we have to apply the subsequent property:
\[\huge \log \left( a \right) + \log \left( b \right) = \log \left( {ab} \right)\]
to this expression:
\[\huge \log \left\{ {{{\left( {x - 2} \right)}^3}} \right\}\log \left( {x + 2} \right) = ...?\]
please continue

Answer 8

oops.. to this expression:
\[\huge \log \left\{ {{{\left( {x - 2} \right)}^3}} \right\} + \log \left( {x + 2} \right) = ...?\]

Answer 9

That would turn into log(x^2-4)^3??

Answer 10

no, we have this:
\[\huge a = {\left( {x - 2} \right)^3},\;b = \left( {x + 2} \right)\]
so, we have:
\[\huge \begin{gathered}
\log \left\{ {{{\left( {x - 2} \right)}^3}} \right\} + \log \left( {x + 2} \right) = \hfill \\
= \log \left\{ {{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)} \right\} \hfill \\
\end{gathered} \]

Answer 11

oh okay, so you wouldn't multiply the bases together?

Answer 12

please wait, since we have to complete the starting expression.
Now I apply this property:
\[\huge \log \left( a \right) - \log \left( b \right) = \log \left( {\frac{a}{b}} \right)\]
to this expression:
\[\Large \log \left\{ {{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)} \right\} - \log \left( {{x^2} - 4} \right)\]
where:
\[\huge a = {\left( {x - 2} \right)^3}\left( {x + 2} \right),\;b = \left( {{x^2} - 4} \right)\]
what do you get?

Answer 13

I got log[(x-2)^3(x+2)] / log(x^2-4)

Answer 14

I got a different result:
\[\Large \begin{gathered}
\log \left\{ {{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)} \right\} - \log \left( {{x^2} - 4} \right) = \hfill \\
\hfill \\
= \log \left\{ {\frac{{{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)}}{{\left( {{x^2} - 4} \right)}}} \right\} \hfill \\
\end{gathered} \]
am I right?

Answer 15

Oh yes, because it is a single logarithm... I have never understood logs, so I'm sorry for all the confusion hehe

Answer 16

ok! :) Now please keep in mind that we have the subsequent identity:
\((x^2-4)=(x-2)(x+2)\), so we can rewrite the last step as below:
\[\Large \begin{gathered}
\log \left\{ {{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)} \right\} - \log \left( {{x^2} - 4} \right) = \hfill \\
\hfill \\
= \log \left\{ {\frac{{{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)}}{{\left( {{x^2} - 4} \right)}}} \right\} = \hfill \\
\hfill \\
= \log \left\{ {\frac{{{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}} \right\} \hfill \\
\end{gathered} \]
now, we can simplify, the expression inside the parenthese, what do you get?

Answer 17

parentheses*

Answer 18

when you have written the answer, please tag

Answer 19

Would it just be log{(x-2)^3 / (x-2)}

Answer 20

@Michele_Laino

Answer 21

such expression, can be simplified further:
so, we have the subsequent steps:
\[\Large \begin{gathered}
\log \left\{ {{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)} \right\} - \log \left( {{x^2} - 4} \right) = \hfill \\
\hfill \\
= \log \left\{ {\frac{{{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)}}{{\left( {{x^2} - 4} \right)}}} \right\} = \hfill \\
\hfill \\
= \log \left\{ {\frac{{{{\left( {x - 2} \right)}^3}\left( {x + 2} \right)}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}} \right\} = \hfill \\
\hfill \\
= \log \left\{ {{{\left( {x - 2} \right)}^2}} \right\} \hfill \\
\end{gathered} \]

Answer 22

hint: