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I want to find the number of terms in the geometric sequence summation, where $a = 5,r = −3,S_n = −910$. After the hard math, I reach $T_n = -1215 = 5(-3)^{n-1}$, which can be solved for $n$.

But this requires the step $\frac{\log{|-243|}}{\log{|-3|}} = n -1$. Usually when you take log of both sides, it is simply $\ln{(x)}$ instead of $\ln{|x|}$. Why can it be done here (even though I know it gives a correct solution).

I am thinking because both the LHS and RHS are the same sign from $T_n = -1215 = 5(-3)^{n-1}$

user71207
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  • The only way this can happen is if $n-1$ is odd and $1215=5 \cdot 3^{n-1}$. This is probably the better way of thinking about it, rather than applying $x \mapsto \ln(|x|)$ (which isn't one-to-one, unlike $\ln$) to both sides. – Ian Aug 04 '21 at 01:39
  • Taking the absolute value doesn't even necessarily work. For example the solution to $-81=(-3)^{n-1}$ is not $n-1=\frac{\log |-81|}{\log |-3|}$. The justification is that we want to find what power we can raise $-3$ to to get $-243$. Since logs don't work well with negative numbers, we will instead work with positive numbers to get $\frac{\log |-243|}{\log |-3|}=5$ and then check that $(-3)^5=-243$ is indeed true. – Alan Abraham Aug 04 '21 at 02:06

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