Expressing as a single logarithm
By Sophia Hammond
I've got the equation:
$$\log_{10}(x^2 - 16) - 3\log_{10}(x + 4) + 2\log_{10} x$$
I'm looking to express this as a single logarithm. I came up with
$$\log_{10}(x^2 - 16) - \log_{10}(x + 4)^3 + \log_{10} x^2$$
then
$$\log_{10} \left(\frac{x^2(x^2 - 16)}{(x + 4)^3}\right) $$
Please forgive me if I got the number of parentheses wrong.
This looks like the results of most of the examples, would you think further simplification is required?
$\endgroup$ 12 Answers
$\begingroup$Your reasoning is absolutely right! However, there is one final simplification you can make: to the fraction $$\frac{x^2(x^2 - 16)}{(x + 4)^3}$$ itself. Hint: Can you factor $x^2-16$?
$\endgroup$ 2 $\begingroup$I'm not sure what you mean when you ask if further simplification is required, but it is possible. Your computations are correct; if you notice that $$(x^2 - 16) = (x+4)(x-4),$$ then you can simplify your last expression, $$\log_{10}\left(\frac{x^2(x^2 - 16)}{(x + 4)^3}\right),$$ to $$\log_{10}\left(\frac{x^2(x - 4)}{(x + 4)^2}\right).$$ Not a major simplification, though.
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