Please help
- Thread starter Sarah123
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topsquark
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If the a's are in geometric progression then they follow the formula [math]a_n = a_0 r^n[/math]. So your three a's will be
[math]a_k = a_0 r^k[/math], [math]a_{k + 1} = a_0 r^{k + 1} = r a_k[/math] and [math]a_0 r^{k + 2} = r^2 a_k[/math] for some k.
What are the logarithms of these numbers?
-Dan
[math]a_k = a_0 r^k[/math], [math]a_{k + 1} = a_0 r^{k + 1} = r a_k[/math] and [math]a_0 r^{k + 2} = r^2 a_k[/math] for some k.
What are the logarithms of these numbers?
-Dan
Thank you. I dont know the logarithms of the numbers, they were not specified in the assignment.
pka
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Decadic logarithms are \(\log_{10}(x)\) called common logarithms.
Dr.Peterson
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You don't need to know any numerical values. Do it symbolically, which is what topsquark suggested.
If a term is [MATH]a_k = a_0 r^k[/MATH], for example, then [MATH]log(a_k) = log(a_0 r^k) = log(a_0) + log(r^k) = log(a_0) + k log(r)[/MATH]. Since [MATH]a_0[/MATH] and [MATH]r[/MATH] are constants, we can think of this as [MATH]log(a_k) = b_0 + k d[/MATH], where [MATH]b_0 = log(a_0)[/MATH] and [MATH]d = log(r)[/MATH]. Does that look familiar?
Or, you could do it a little differently. If the numbers form a geometric sequence, then their ratios are the same. For the logs to form an arithmetic sequence, their differences have to be the same. What is the differences of the logs of two numbers whose ratio is r (as an expression in terms of r)?
topsquark
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Let me give an example of what's needed here. [math]log ( x y^3) = log(x) + log(y^3) = log(x) + 3 log(y)[/math].
Do that for all three values and see if they fit into an arithmetic series.
You don't need the actual numbers.
-Dan