Ideally, we would treat e^x and ln(x) = log~e(x) as the default. We know that a^x = e^(x ln a), and log~a(x) = ln x / ln a. So if we introduced an operator ^(x)

Ideally, we would treat e^x and ln(x) = log~e(x) as the default. We know that a^x = e^(x ln a), and log~a(x) = ln x / ln a. So if we introduced an operator ^(x) = e^x, and another v(x) = ln(x), we could write a^x = ^xva, a√x = ^(vx/a), log~a(x) = vx/va. Common identities would be written thus: ^xva * ^yva = ^(xva + yva) = ^(x + y)va ^yv(^xva) = ^yxva (note that this is just the identity v(^x) = x) ln(a^x) = x ln a is just v(^xva) = xva ^yvx = z <=> yvx = vz <=> y = vz/vx ^yvx = z <=> yvx = vz <=> vx = vz/y <=> x = ^(vz/y) (alternate derivation:) ^yvx = z <=> ^(v^yvx/y) = ^(vz/y) <=> x = ^(vz/y) Differentials are thus: D(^f) = Df^f D(vf) = Df/f d/dx(^xva) = va^xva d/dx(^nvx) = d/dx(nvx)^nvx = n/x * ^nvx = n ^-1vx ^nvx = n^(n-1)vx d/dx(vx/va) = 1/xva