A couple things.
Wolfram Alpha is very helpful in fixing any syntax errors, while Mathematica is not.
You have a couple of typos in the equations you typed in. using kM instead of k M or k*M -- what you have is the single variable kM, instead of the multiplied variables k * M.
Also, I would not use capital letters as variable names, because there might be a conflict with a built-in Mathematica symbol. You also clear the symbol La, while the equation uses L.
I also defined the constants before using Solve[], since I got an error otherwise.
Note that you did this substitution manually when posing the problem to W|A.
In[13]:= ClearAll[k, f, m, l, d]
In[14]:= f = 0.02
l = 12.0
d = 0.02
k = 1.4
Out[14]= 0.02
Out[15]= 12.
Out[16]= 0.02
Out[17]= 1.4
In[18]:= Solve[(4 f l/
d) == ((1 -
m^2)/(k m^2)) + (((k +
1)/(2 k)) (Log[((k + 1) m^2)/(2 + ((k - 1)*m^2))])), m]
Mathematica is busy trying to solve the equation as I type this, and gives no error messages. (There may be no symbolic solution, which is what you asked for with Solve[].)
I would try something besides Solve[], such as Solve[], or FindRoot[]. I suspect that W|A does the same thing in order to give a numerical value so quickly.
Read the documentation.