How to solve: ln 2x+3(ln x-ln y)

Question

I do not understand how to solve logarithmic equations. The topic is very confusing to me. I really need help.

Answer 1

the basic thing to understand about logarithms is that they are basically reverse exponents..

Log_ab  basically asks "To what power do I raise a to get b?"

 

Log_ab = c   means that  a^c = b

 

So if we have xⁿ = y      then  log_xy=n

 

ln is   log_e    e is a number approximately equal to 2.718282..     e is Euler's number...

At the moment it isn't really important to know where "e" comes from but

realize ln is log_e   so   ln(e)=1 because it asks  "to what power do I raise e to get e?" ... 1

 

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ln(2x) + 3(ln(x) - ln(y))

 

Note:  log_ax - log_ay = log_a(x/y) so  lnx - lny = ln(x/y)

Then we have..

 

ln(2x) + 3ln(x/y)

 

Note:  n(log_ax) = log_axⁿ so  3(ln(x/y)) = ln(x/y)³ so we have:

 

ln(2x) + ln (x/y)³

 

Note:  log_ax + log_ay = log_axy   so  we have:

 

ln((2x)(x/y)³) = ln (2x⁴/y³)

 

 

 

This combines the problem to a single form ... but does not actually solve an equation.

In order to solve the problem to get a numeric value of x and y we would need to have

two equations so we could use substitution or elimination.