I know that log has this product rule:
$$\ln (9\cdot x)=\ln 9+\ln x$$
then what about $\ln(9 + x)$? Could that be simplified further?
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4No. It's not possible in general simplify $\ln(a + b)$. – jjagmath Feb 13 '22 at 01:08
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To gain an intuition as to why this is true, note that $e^9e^x$ simplifies to $e^{9+x}$ but what does $e^9+e^x$ simplify to? – morrowmh Feb 13 '22 at 01:13
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$$\ln \max(x, 9) \le \ln (9+ x) \le \ln \max(x, 9) + \ln 2$$ (for $x \ge 1$ anyway) – DanielV Feb 13 '22 at 01:21
1 Answers
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These laws of logarithm are derived from the laws of exponents, since there are no laws of exponents when two are added, these, in general, cannot be simplified more. And we usually write the answer as $$ln(a+b)$$ in such cases.
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