SOLUTION: Given 𝑙𝑜𝑔5 2 = 𝑚 and 𝑙𝑜𝑔5 8 = 𝑛, express 𝑙𝑜𝑔5 6.4 in terms of m and n.

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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): You can put this solution on YOUR website!

Log rules
where b > 0 and
Also x > 0 and y > 0.
The order of the rules does not matter. They are placed into a numbered order so I can reference them down below.





=

= Use rule 3 shown above

=

= Use rule 1 and rule 2.

= Substitute in m & n; use rule 4

=

Therefore,
when and

--------------------------------------------------------------------------

Notice the following


Triple both sides

Apply rule 1 shown above





So,


It does seem a bit strange how we can write the expression in terms of just one variable, even though your teacher says "in terms of m and n" implying your teacher wants both variables present.
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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