Let z = (bx - 1)/x
Multiply both sides by x and add 1 to both sides of the result:
1 + zx = bx
Take (natural) logs of both sides
ln(1+zx) = xln(b) using the property of logs that ln(mn) = n ln(m)
Divide both sides by x
(1/x)*ln(1 + zx) = ln(b)
Use the same property of logs as we used above, but in reverse:
ln( (1 + zx)1/x ) = ln(b)
By definition, ez = limit as x goes to zero of (1 + zx)1/x
So in the limit as x goes to zero we have
ln(ez) = ln(b)
or just z = ln(b)
or, using the definition of z
(bx - 1)/x = ln(b)
Let z = (bx - 1)/x
Multiply both sides by x and add 1 to both sides of the result:
1 + zx = bx
Take (natural) logs of both sides
ln(1+zx) = xln(b) using the property of logs that ln(mn) = n ln(m)
Divide both sides by x
(1/x)*ln(1 + zx) = ln(b)
Use the same property of logs as we used above, but in reverse:
ln( (1 + zx)1/x ) = ln(b)
By definition, ez = limit as x goes to zero of (1 + zx)1/x
So in the limit as x goes to zero we have
ln(ez) = ln(b)
or just z = ln(b)
or, using the definition of z
(bx - 1)/x = ln(b)