Solving An Interesting Log Equation | Math Olympiads

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[spelunkerd]

What I like about your videos is how you consistently do not choose the obvious pathway that most of us would use. In this case rather than following the blaring siren to change base, you chose more elegant pathways to get the same place. Just because they may take a little longer at times does not take away the theme that there are often many options. Along the way we recall principles that aren't necessarily used every day.

[justabunga]

There is another method that you can also use, which is the change of base formula. This equation can be rewritten as Multiply everything by the LCD, which is log(2)log(3), and factor out log(x). This means that Divide both sides by log(3)+log(2), or log(6). log(x)=log(2)log(3)/log(6), and x=10^(log(2)log(3)/log(6)). Simplify this even better and using properties of logarithms, which is x=2^(log_6(3)). The answer can also be x=3^(log_6(2)) if we interchange the the base of the exponent and the log of an argument.

[erikdahlen]

I did log base change in my head when i saw the problem, factored, and solved for x. easier than whatever you did here

[Education-]

My answer is same but another approach

First i use reciprocal property
To make same base
And then take LCM
And apply log(a) + log(b) = log(ab)
After that
Use base changing property
And I got answer 2




JEE aspirant
2025
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[rajatdogra]

Log x ( log 2 + log 3) = log 2 * log 3

[christianlopez]

log2x=lnx/ln2, log3x=lnx/ln3....easyly you can find lnx.

[MushookieMan]

I just used the change of base formula on one of the logs, wasn't a bad method

[MarCamus]

Changed both bases to ln
and after some algebra I got the solution in terms of ln and e

[MrGeorge]

I suggest using ln(b) / ln(a) instead of log_a(b).

ln(x) (1 / ln(2) +1 / ln(3)) = 1
ln(x) = ln(2) * ln(3) / ln(6)

so x = e ^ (ln(2) * ln(3) / ln(6)) = 1.52959...

[somnathkundu]

Do the change of base and add the terms.

[moeberry]

The fastest way is to use change of bases. So you obtain log(x)/2+log(x)/3. Now just factor out a log(x) and divide by that constant on both sides and then anti log both sides and your done.

[KookyPiranha]

Aint no way this is a math olympiad problem. I saw a similar one in my hw

[tetsuyaikeda]

I could know 'Zee is not see, nor c, nor be ' that must be some kind of joking

[scottleung]

I got the same answer the first method had.

[popitripodi]

I used the change of base method!!your approach was very good alternative!!!!❤❤❤❤

[GreenMeansGOF]

change of base is how I would have done it

[phongvong]

It is over I mean they are. Then something has to be prioritised. Please. Thank you.

[gatosimple]

Profe menos palabras demás, sea más directo

[MichaelRothwell]

In case anyone is wondering, the reason for the rule
a^(log꜀b)=b^(log꜀a)
is that if you log each side to base c and use the power rule for logs, both sides give log꜀a log꜀b.
Putting it another way,
c^(log꜀a
and
c^(log꜀b
so a^(log꜀b)=b^(log꜀a).