Verified answer

The logarithm equation you presented is the change of base law.

C can be any number, commonly 10 or e as our calculators only work in bases of 10 or e and this law helps to solve log equations of other bases.

e is the natural logarithm number (its like the concept of pi, e= approx. 2.71....)

e is a special number defined as follows-

e = 1+ 1/1! + 1/2! +-----------to infinity

value of e is approximately 2.7.....

if a=e^x then x = log a(to the base e)

if a^x = b then we say log b(to the base a)=x

let a^x=b & c^y=a

so, y=log a(to the base c)

now b= a^x = (c^y)^x = c^(xy)

so, log b(to the base c) =xy = log b(to the base a) * log a(to the base c)

hence

log b(to the base a) = log b(to the base c)/ log a(to the base c)

that is the formula for converting a logarithm of a number in one base to the logarithm of another base.

e is a constant. Its value is 2.7... It has the special property

that the derivative of the function e^x is itself.

log(b,a) = log(b,c)/log(a,c)

The above formula is always true. So a,b,c can be any positive number.

Basically e to the power of log(a,e) = a or e^log(a,e) = a

a^log(b,a) = b

c^log(a,c) = a

c^log(b,c)=b

Now,

a^log(b,a) = b

so

(c^log(a,c))^log(b,a)=b

so

c^(log(a,c)log(b,a))=b

so (log(b,c)=log(a,c)log(b,a)

So log(b,a)=log(b,c)/log(a,c)

the first one: it means a=b=c

second one: e is a number that was used to oftern (approx...2.178.......) so it was named "e"...log e = ln

third one: ln a