Question 1206465: Given 𝑙𝑜𝑔5 2 = 𝑚 and 𝑙𝑜𝑔5 8 = 𝑛, express 𝑙𝑜𝑔5 6.4 in terms of m and n.
Found 4 solutions by Theo, ikleyn, math_tutor2020, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you are given that log5(2) = m and log5(8) = n.
since 8^2 / 10 = 64 / 10 = 6.4, your equation is:
log5(6.4) = log5(8^2 / 10) = log5(8^2) - log5(10).
since log5(8^2) = 2 * log5(8), then log5(6.4) = 2 * log5(8) - log5(10).
log5(10) = log5(2 * 5) = log5(2) + log5(5).
since log5(5) = 1, then log5(10) = log5(2) + 1.
put them together and you get log5(6.4) = 2 * log5(8) - (log5(2) + 1)
simplify to get log5(6.4) = 2 * log5(8) - log5(2) - 1.
since log5(8) = n and log5(2) = m, this becomes log5(6.4) = 2 * n - m - 1
that's your solution, far as i can tell.
to see if that's true, we can use the log base conversion formula of log5(x) = log(x) / log(5).
that gets us log5(6.4) = log(6.4) / log(5) = 1.15338279.
note that log(x) is the log function of your calculator, which is programmed to give you log10(x).
the log base conversion formula is really telling you that log5(x) = log10(x)/log10(5).
we drop the log10 by convention and just show log(x)/log(5).
when you use the log function of your calculator, you are providing log10(x), which means log of x to the bae of 10.
the conversion formula allows us to provide the log of any base, using the calculator.
that's why, when we want log5(6.4), we simply take log(6.4) / log(5) and we get it.
anyway, back to your problem.
by log rules, log5(6.4) = 1.15338279 if and only if 5^1.15338279 = 6.4
we use our calculator to confirm that's true.
it is.
to confirm that log5(6.4) = 2 * n - m - 1, we replace n with log5(8) and m with log5(2) to get:
log5(6.4) = 2 * log5(8) - log5(2) - 1.
that becomes log5(6.4) = 2 * log(8)/log(5) - log(2)/log(5) - 1.
that becomes log5(6.4) = 1.15338279.
we already confirmed earlier that 5^1.15338279 = 6.4, so that's true.
your solution, as best i can determine, is that log5(6.4) = 2 * n - m - 1
Answer by ikleyn(52908)  (Show Source):
You can put this solution on YOUR website! .
This condition " Given  and  " looks strange,
because if is given, then  is automatically known, too.
In other words, then m and n are not independent constant values, because n = 3m.
Therefore, the condition of the problem, as it is given in the post, looks more than strange to me,
revealing a complete mathematical illiteracy of its creator.
To become a TRUE Math problem, this post must be edited/re-written from scratch.
Otherwise, the readers will laugh on it.
If a person does not see it, it means that the person is mathematically blind.
. . . . . . . .
When a student is mathematically blind, it is normal - there is no tragedy in it . . .
But when a person, positioning himself as a Math composer, is mathematically blind,
it is just a SHAME.
Answer by math_tutor2020(3817)  (Show Source):
Answer by greenestamps(13215)  (Show Source):
You can put this solution on YOUR website!
The response from tutor @ikleyn is unwarranted. Yes, from the given information we can conclude that n = 3m.
However, the instruction to express log base 5 of 6.4 in terms of m and n is still a valid exercise for the student who is learning how to work with logarithms. Writing the problem that way does not demonstrate mathematical illiteracy.
The response from tutor @Theo is fine; but with all the words he puts in his responses it is often hard to find the actual solution.
Here is a more concise solution.
Expressing 6.4 using factors of 8, 2, and the base 5, we have
So, using basic rules of logarithms,
ANSWER: 2n-m-1
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