# Drill 3 Natural log and probability

Subject: Deep Learning

Exam: MT2

Reader: https://deep-learning-study-room.vercel.app/#mt2/read/17/mt2-c17-s3

Canonical: https://deep-learning-study-room.vercel.app/read/dl/mt2/section/mt2-c17-s3/

## Question

**Question.** Compute −ln(0.9),−ln(0.1), and  $e^{-\ln 2}$ .

## Answer — step by step

### Given

Probabilities: 0.9 and 0.1.

Exponential expression: exp(−ln 2).

### To find

Find −ln(0.9), −ln(0.1) and exp(−ln 2).

### Formula — which one and why

**Natural logarithms.** The natural logarithm reverses the exponential. A negative log turns a small correct-class probability into a larger positive loss.

ln uses base e ≈ 2.71828. exp(a) means e raised to the power a. Keep the minus outside ln when the question says −ln(p).



$$
-\ln p=\ln\frac1p
$$





$$
e^{\ln a}=a
$$





$$
e^{-\ln a}=\frac1a
$$



**Yaad rakhne ka cue:** exp banata hai power; ln us power ko wapas nikaalta hai.

### Solution — step by step

### Meaning and formulas

The natural logarithm ln tells us the power of e needed to obtain a number. It reverses the exponential operation. A calculator evaluates ln; it is not ordinary division.



$$
\ln(e^a)=a,\qquad e^{\ln a}=a\quad(a>0)
$$





$$
e^{-a}=\frac1{e^a}
$$



### Step 1: calculate the first logarithm, then its minus sign



$$
\ln(0.9)\approx-0.105360516
$$





$$
-\ln(0.9)=-(-0.105360516)=0.105360516
$$



### Step 2: calculate the second logarithm



$$
\ln(0.1)\approx-2.302585093
$$





$$
-\ln(0.1)=-(-2.302585093)=2.302585093
$$



When these are correct-class probabilities, 0.1 represents a worse prediction than 0.9, so its negative-log loss is larger.

### Step 3: simplify the exponential using the inverse rule



$$
e^{-\ln2}=\frac1{e^{\ln2}}=\frac12=0.5
$$



**Final answers:** 0.105360516, 2.302585093 and 0.5. Use the ln key; base-10 log gives different first and second answers.