---
title: "Output of three log amps with summer"
subject: "Advanced Electronics"
chapter: 6
key: "semester_4/advanced_electronics:three-log-amps-summing"
source: https://www.bctengineer.com/q/semester_4/advanced_electronics/three-log-amps-summing
site: BCT Engineering
---

# Output of three log amps with summer

Advanced Electronics · Chapter 6: Log-Antilog Circuit Applications

- Asked in: 2071 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/advanced_electronics/three-log-amps-summing
- Whole chapter: https://www.bctengineer.com/md/semester_4/advanced_electronics/chapter-6

## Question

Find the relationship of input and output voltage in the following figure. [Figure: three log amplifiers: V1, V2 and V3 each pass through a resistor R to the inverting input of their own op-amp (non-inverting inputs grounded), with transistors Q1, Q2 and Q3 (bases grounded) as the respective feedback elements. Each log-amp output passes through a resistor R to the inverting input of a summing op-amp with feedback resistor 2R; its non-inverting input is grounded; its output is V0.]

## Short answer

Each stage is a log amp giving V0i = -VT ln(Vi/(R Is)). The summing amplifier with feedback 2R and input resistances R has gain -2, so V0 = 2VT ln(V1 V2 V3/(R Is)^3). The output is proportional to the log of the product cubed.

## Answer

Each input stage is a **log amplifier** (transistor in the feedback path), and the last stage is an **inverting summer** with gain −2R/R = −2.

#### Log amplifier outputs

For a log amp with input resistor R and grounded-base transistor in feedback:

```text
V01 = −V_T ln( V1 / (R·Is) )
V02 = −V_T ln( V2 / (R·Is) )
V03 = −V_T ln( V3 / (R·Is) )
```

where V_T = kT/q ≈ 26 mV at room temperature and Is is the reverse saturation current (transistors assumed matched).

#### Summing amplifier

Inputs through R, feedback 2R:

```text
V0 = −(2R/R)(V01 + V02 + V03)
   = −2 (V01 + V02 + V03)
```

Substituting:

```text
V0 = 2V_T [ ln(V1/RIs) + ln(V2/RIs) + ln(V3/RIs) ]
   = 2V_T ln( V1·V2·V3 / (R·Is)³ )
```

**Answer: V0 = 2V_T · ln[ V1·V2·V3 / (R·Is)³ ]**

This can also be written as V0 = V_T ln[ (V1·V2·V3)² / (R·Is)⁶ ], i.e. the output is proportional to the log of the square of the product of the three inputs. The circuit works only for positive V1, V2, V3.

## Check yourself

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

1. What is the output of each log stage?
   A. V0n = -VT ln(Vn/(R Is))
   B. V0n = VT exp(Vn)
   C. V0n = -R Is Vn
   D. V0n = VT ln Vn R
   Answer: A. It is the basic log amp with input resistor R. (Key point: Log amp output)

2. What is the summer gain?
   A. -2, since feedback 2R and input R
   B. -1
   C. +2
   D. -1/2
   Answer: A. Gain = -2R/R = -2. (Key point: Summer gain)

3. What is V0?
   A. VT ln[V1 + V2 + V3]
   B. -2 VT ln[V1 V2 V3]
   C. 2 VT ln[V1 V2 V3/(R Is)]
   D. 2 VT ln[V1 V2 V3/(R Is)^3]
   Answer: D. Summing three logs gives the log of a product, with (R Is)^3 in the denominator. (Key point: Final result)

## Related in Advanced Electronics

- [Output voltage of two op-amp IA](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Atwo-opamp-ia-output) — Advanced Electronics, chapter 4
- [Output of log-ratio amplifier with RT compensation](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Alog-ratio-temperature-compensated) — Advanced Electronics, chapter 5
- [Find coefficients of op-amp summing circuit](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Asumming-amplifier-coefficients) — Advanced Electronics, chapter 1
- [Input impedance of inverting op-amp](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ainverting-input-impedance) — Advanced Electronics, chapter 2
- [Output offset voltage of 741 inverting amplifier](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Aoutput-offset-741-inverting) — Advanced Electronics, chapter 2
- [Log-antilog circuit: Vo in terms of Vi and VR](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Alog-antilog-circuit-vo-vi) — Advanced Electronics, chapter 5, asked 3×

## Related in other subjects

- [Voltage gain of non-inverting op-amp](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 2×
- [Non-inverting op-amp voltage gain](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Anon-inverting-amplifier-gain) — Fundamental of Electrical and Electronics Engineering, chapter 6
- [Ideal op-amp properties; non-inverting gain](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 3×
- [Ideal op-amp properties; inverting gain](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-inverting-gain) — Basic Electronics Engineering, chapter 4, asked 3×