---
title: "Check linearity/TI: log(x(t)), t·x(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:check-log-xt-and-t-xt"
source: https://www.bctengineer.com/q/semester_5/data_communication/check-log-xt-and-t-xt
site: BCT Engineering
---

# Check linearity/TI: log(x(t)), t·x(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2070 Asar, 6 marks
- Video: [Linear and Non-Linear Systems (Solved Problems) | Part 1](https://www.youtube.com/watch?v=QcAY1oPozhw) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/check-log-xt-and-t-xt
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Check whether the given system is linear and time invariant or not. i) y(t) = log(x(t)) ii) y(t) = t·x(t)

## Short answer

i) y=log(x(t)): non-linear (log not additive) but time-invariant. ii) y=t·x(t): linear (amplification by t is linear) but time-variant because delay alters multiplier. Neither system is LTI.

## Answer

#### i) y(t) = log(x(t))
- **Linearity**: for x = a·x₁ + b·x₂, y = log(a·x₁ + b·x₂), while a·y₁ + b·y₂ = a·log x₁ + b·log x₂. They are not equal (log is not additive), and log(a·x) = log a + log x ≠ a·log x. **Non-linear.**
- **Time invariance**: input x(t - t₀) gives log x(t - t₀) = y(t - t₀). **Time-invariant.**

#### ii) y(t) = t·x(t)
- **Linearity**: response to a·x₁ + b·x₂ is t(a·x₁ + b·x₂) = a·t·x₁ + b·t·x₂ = a·y₁ + b·y₂. **Linear.**
- **Time invariance**: delay the input: y₁(t) = t·x(t - t₀). Delay the output: y(t - t₀) = (t - t₀)·x(t - t₀). These differ. **Time-variant.**

**Answer: (i) non-linear and time-invariant; (ii) linear but time-variant. Neither system is LTI.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Is y = log x(t) linear?
   A. Yes
   B. Only if x > 0
   C. Yes if x is real
   D. No, log is not additive
   Answer: D. log(a x1 + b x2) differs from a log x1 + b log x2. (Key point: log linearity)

2. Is y = log x(t) time-invariant?
   A. Yes, a shifted input gives the shifted output
   B. No, log depends on t
   C. No, it is nonlinear
   D. Cannot tell
   Answer: A. No explicit t dependence. (Key point: log time invariance)

3. Is y = t x(t) linear?
   A. No
   B. Yes, t is a fixed multiplier
   C. Only for t>0
   D. No, since it varies with t
   Answer: B. Superposition holds. (Key point: t x(t) linearity)

4. Why is y = t x(t) time-variant?
   A. It is nonlinear
   B. It has memory
   C. Delayed input gives t x(t-t0) but delayed output gives (t-t0)x(t-t0)
   D. It is unstable
   Answer: C. The explicit t factor breaks time invariance. (Key point: t x(t) time invariance)

## Related in Data Communication

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