The simplification of 3log(x + 4) – 2log(x – 7) + 5log(x - 2) - log(x^2) is [tex]\log \left(\frac{(x+4)^{3} \times(x-2)^{5}}{(x-7)^{2} \times x^{2}}\right)[/tex]
Solution:
Given, expression is [tex]3 \log (x+4)-2 \log (x-7)+5 \log (x-2)-\log \left(x^{2}\right)[/tex]
We have to write in as single logarithm by simplifying it.
Now, take the given expression.
[tex]\rightarrow 3 \log (x+4)-2 \log (x-7)+5 \log (x-2)-\log \left(x^{2}\right)[/tex]
Rearranging the terms we get,
[tex]\left.\rightarrow 3 \log (x+4)+5 \log (x-2)-2 \log (x-7)+\log \left(x^{2}\right)\right)[/tex]
[tex]\text { since a } \times \log b=\log \left(b^{a}\right)[/tex]
[tex]\rightarrow \log (x+4)^{3}+\log (x-2)^{5}-\left(\log (x-7)^{2}+\log \left(x^{2}\right)\right)[/tex]
[tex]\text { We know that } \log a \times \log b=\log a b[/tex]
[tex]\rightarrow \log \left((x+4)^{3} \times(x-2)^{5}\right)-\left(\log \left((x-7)^{2} \times\left(x^{2}\right)\right)\right.[/tex]
[tex]\text { We know that } \log a-\log b=\log \frac{a}{b}[/tex]
[tex]\rightarrow \log \left(\frac{(x+4)^{3} \times(x-2)^{5}}{(x-7)^{2} \times x^{2}}\right)[/tex]
Hence, the simplified form [tex]\rightarrow \log \left(\frac{(x+4)^{3} \times(x-2)^{5}}{(x-7)^{2} \times x^{2}}\right)[/tex]
