Question

Simplify the expression below and write it as a single logarithm:

3log(x + 4) – 2log(x – 7) + 5log(x - 2) - log(x2)

Answer 1

other person is wrong

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Answer 2

The simplification of 3log(x + 4) – 2log(x – 7) + 5log(x - 2) - log(x^2) is
`\log \left(((x+4)^(3) *(x-2)^(5))/((x-7)^(2) * x^(2))\right)`

Solution:

Given, expression is
`3 \log (x+4)-2 \log (x-7)+5 \log (x-2)-\log \left(x^(2)\right)`

We have to write in as single logarithm by simplifying it.

Now, take the given expression.

`\rightarrow 3 \log (x+4)-2 \log (x-7)+5 \log (x-2)-\log \left(x^(2)\right)`

Rearranging the terms we get,

`\left.\rightarrow 3 \log (x+4)+5 \log (x-2)-2 \log (x-7)+\log \left(x^(2)\right)\right)`

`\text { since a } * \log b=\log \left(b^(a)\right)`

`\rightarrow \log (x+4)^(3)+\log (x-2)^(5)-\left(\log (x-7)^(2)+\log \left(x^(2)\right)\right)`

`\text { We know that } \log a * \log b=\log a b`

`\rightarrow \log \left((x+4)^(3) *(x-2)^(5)\right)-\left(\log \left((x-7)^(2) *\left(x^(2)\right)\right)\right.`

`\text { We know that } \log a-\log b=\log (a)/(b)`

`\rightarrow \log \left(((x+4)^(3) *(x-2)^(5))/((x-7)^(2) * x^(2))\right)`

Hence, the simplified form
`\rightarrow \log \left(((x+4)^(3) *(x-2)^(5))/((x-7)^(2) * x^(2))\right)`