logB(A) asks what exponent the base B needs to become A?

log5(25) asks what exponent the base 5 needs to become 25?

The answer is 2 because when 5 is raised to the 2 power the result is
25.  Therefore 2 is the exponent the base 5 needs to become 25.
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log2(16) asks what exponent the base 2 needs to become 16?
 
The answer is 4 because when 2 is raised to the 4th power the result is
16.  Therefore 4 is the exponent the base 2 needs to become 16.
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log10(1000) asks what exponent the base 10 needs to become 1000?
 
The answer is 3 because when 10 is raised to the 3rd power the result is
1000.  Therefore 3 is the exponent the base 10 needs to become 1000.

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e is an irrational number which is approximately: 

2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746639193200305992...

It is a special number that has the property that for any
point on the graph of y=e^x, the slope of the line tangent
to the graph at that point is the same as the y-coordinate
of that point. 

ln(x) means the logarithm of x to the base e  ln(x) = loge(x)

If log(x) is written without a base, the base is understood t6o be 10.

So log(x) means log(x)

Edwin