# Exponentials and natural logarithms

Subject: Deep Learning

Exam: MT2

Reader: https://deep-learning-study-room.vercel.app/#mt2/read/2/mt2-c2-s6

Canonical: https://deep-learning-study-room.vercel.app/read/dl/mt2/section/mt2-c2-s6/

**Definitions.** The exponential function exp(x) = eˣ uses Euler's constant e. The natural logarithm ln is its inverse on positive real values. These functions occur in sigmoid, tanh, softmax and cross-entropy.



$$
e^{-x}=\frac1{e^x},\qquad\ln(e^x)=x
$$





$$
e^{\ln x}=x\ (x>0),\qquad\ln(ab)=\ln a+\ln b\ (a,b>0)
$$



For positive constants and valid domains, exponent and log rules simplify expressions before a calculator is needed. For example, e^(−ln 2) = 1/2. Use natural ln for the loss formulas, not the calculator's base-10 log.

### Derivatives used in neural networks



$$
\frac{d}{dz}e^{-z}=-e^{-z},\qquad\frac{d}{dp}[-\ln p]=-\frac1p
$$



The minus sign in the first expression comes from differentiating the inner function −z. Applying the chain rule correctly here gives the sigmoid derivative derived in the activation section.

### Brief integration and trigonometry reminder

Integration reverses differentiation up to an additive constant. The basic relevant identities are:



$$
\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)
$$





$$
\int\frac1x\,dx=\ln|x|+C,\qquad\int e^x\,dx=e^x+C
$$



For angles in radians, d(sin x)/dx = cos x and d(cos x)/dx = −sin x. The identity sin²x + cos²x = 1 concerns ordinary trigonometric functions. Hyperbolic tanh is a different function; using tan instead of tanh gives an incorrect recurrent-network answer.



$$
\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}},\qquad\frac{d}{dz}\tanh z=1-\tanh^2z
$$



The supplied MT2 calculation families mainly require derivatives, matrix products, logarithms and elementary statistics. Extended integration or trigonometric exercises are not needed for those families.

### Calculator values for activation calculations

At z = 2, sigmoid = 1/(1 + e⁻²) ≈ 0.880797. At z = 1, tanh ≈ 0.761594. Keep unrounded values when a later gradient uses these activations.