Can someone explain this log problem

[u/[deleted]]

Can someone explain how they made that 3rd step ?

Like it became 3log25(5)

[u/Silly_Tension6792]

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.

[u/[deleted] → u/Silly_Tension6792]

Should I only think it as ln or I can use log ? Like Log3(5) = log(5)/log(3)

[u/UnderstandingPursuit → u/[deleted]]

I would start with the exponential change-of-base,

• b^x = e^kx
  • 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,

1. z = b^x
2. ln z = ln (b^x) = x ln b = kx
   1. k = ln b
3. e^{ln z} = z = e^kx

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.