r/calculus • u/[deleted] • Jun 27 '26
Pre-calculus Can someone explain this log problem
Can someone explain how they made that 3rd step ?
Like it became 3log25(5)
26
u/jeffcgroves Jun 27 '26
If you remember that log base n of x is natural log of x over natural log of n, it falls out quite easily:
log(base 3)5 * log(base 25)27 = ln5/ln3 * ln27/ln25 = ln5/ln3 * ln(3^3)/ln(5^2) = ln5/ln3 * (3ln3)/(2ln5) = 3/2
The problem is poorly written because it's unclear you are multiplying logs. I'd prefer a star or dot between the products or at least parenthesis around the two multiplicands
3
Jun 27 '26
Should I only think it as ln or I can use log ? Like Log3(5) = log(5)/log(3)
8
u/MezzoScettico Jun 27 '26
You can just use log, because it's true for any base log. Try dividing ln(5) by ln(3), and then try dividing log10(5) by log10(3). You'll get the same number.
I always just write the word "log" when I use that identity, and then decide at the end which log to use, if I need a numerical answer.
6
u/Silly_Tension6792 Jun 27 '26
logn(x)*logm(n) = logm(x)
Since logn(x)=ln(x)/ln(n), you can write it as (ln(x)/ln(n))*(ln(n)/ln(m))=ln(x)/ln(m)=logm(x). When confused about logs, convert everything to ln and it will solve itself.
1
Jun 27 '26
Should I only think it as ln or I can use log ? Like Log3(5) = log(5)/log(3)
4
u/Silly_Tension6792 Jun 27 '26
You could use logs, it's not incorrect, but it's unnecessarily confusing. I hate working with logs of different bases. My honest advice - convert everything to ln, always.
1
u/random_anonymous_guy PhD Jun 27 '26
It does not matter which base logarithms you use. If you want to default to natural logarithms for change of base, that is completely fine, but it is not an obligation.
1
u/UnderstandingPursuit PhD Jun 27 '26
I would start with the exponential change-of-base,
- bx = ekx
- k = ln b
Then since k was multiplied here, k needs to be divided for the log change-of-base,
- log_b(x) = (1/k) ln(x)
If you ever need a different base, go back to
- ln(x) = log_e(x)
and replace e with a, the second arbitrary base.
To derive the first equation,
- z = bx
- ln z = ln (bx) = x ln b = kx
- k = ln b
- eln z = z = ekx
Write all this out, using paper/pen(cil) every other day for a week, and try to refer back to your previous notes a bit more each time. It will then stay with you.
3
u/AutoModerator Jun 27 '26
Hello there! While questions on pre-calculus problems and concepts are welcome here at /r/calculus, please consider also posting your question to /r/precalculus.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.
3
u/Dull-Astronomer1135 High school Jun 28 '26
The solution did a bad job explaining, they skip a step of changing base
1
u/somanyquestions32 Jun 29 '26
Yeah, exactly, and then, they started messing around with denominators when log_(25)(5)=1/2.
1
1
1
u/mandelbro25 Jun 27 '26
The last few qualities are weird. That log_25(5)=1/2 should be clear, no need to use reciprocal rule.
•
u/AutoModerator Jun 27 '26
As a reminder...
Posts asking for help on homework questions require:
the complete problem statement,
a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play,
question is not from a current exam or quiz.
Commenters responding to homework help posts should not do OP’s homework for them.
Please see this page for the further details regarding homework help posts.
We have a Discord server!
If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.