The operational amplifier circuit configurations which can perform mathematical operations such as log and antilog (exponential), including an amplification of the input signal provided to the circuit, are known as Logarithmic amplifier and Antilogarithmic amplifier respectively. In this section, we are going to learn about the Logarithmic amplifier and Antilog in detail.

**Contents:**

**Introduction****Logarithmic (Log) Amplifier****Log amplifier configuration****Diode based Log amplifier configuration****Transistor based Log amplifier****configuration****Output and Working Principle of Log Amplifier****Applications of the log amplifier****What is Antilog?****Antilog Amplifier****Log amplifier configuration****Diode based antilog amplifier configuration****Transistor based antilog amplifier configuration****Output and Working Principle of Log Amplifier****Applications of the antilog amplifier**

**Logarithm (Log) Amplifier**

An operational amplifier in which the output voltage of the amplifier (V_{0}) is directly proportional to the natural logarithm of the input voltage (V_{i}) is known as a logarithmic amplifier. Basically, the natural logarithm of the input voltage is multiplied by a constant value and produced as output.

**Log Amplifier Circuit**

**Log Amplifier Using Transistor**

**Log Amplifier using Diode**

**Output and Working Principle of Log Amplifier**

This can be expressed as follows:

Where K is the constant term, and V_{ref} refers to a normalization constant, which we get to know in this section.

Generally, logarithm amplifiers may require more than one op-amp, in which case they are known as compensated logarithm amplifiers. They even require high performing op-amps for their proper functioning, such as LM1458, LM771, and LM714, are being some of the widely used logarithm amplifier.

The diode is connected in forward biasing. So, the diode current can be represented as:

Where I_{s} is the saturation current, V_{D} is the voltage drop for the diode. The V_{T} is the thermal voltage. The diode current can be rewritten with high biasing condition,

The i_{1} expressed by,

Since the voltage at inverting terminal of the op-amp is at virtual ground, hence, the output voltage is given by V_{0 }= -V_{D}

Noting that i_{1 }= i_{D}, we can write

But, as noted earlier, V_{D }= -V_{0} and so,

Taking natural logarithm on both sides of this equation, we found

Or,

The equation of the output voltage (V_{0}) of the logarithm amplifier contains a negative sign, which indicates that there is a phase difference of 180 ^{o}. Or,

A more advanced one utilize bipolar transistors to remove I_{s} in the logarithmic term. In this type of logarithm amplifier configuration, the output voltage is given as:

**Applications of the logarithmic amplifier**

Log amplifier is used for mathematical applications and also in different devices as per their need. Some of the applications of the log amplifier are as follows:

- Log amplifiers are used for mathematical applications, mainly in multiplication. It is also used in the division and other exponential operations too. As it can perform multiplication operation, hence it is used in analog computers, in synthesizing audio effects, measuring instruments that require multiplication operation such as in calculating power (multiplication of current and voltage).
- As we know that when we need to calculate the decibel equivalent of a given quantity, we require the use of a logarithmic operator, and hence, log amplifiers are used to calculate decibel (dB) value of a quantity.
- Monolithic logarithmic amplifiers are used in certain situations, like in Radio Frequency domain, for efficient spacing (reducing components and space needed by them), and also to improve bandwidth and noise rejection.
- It is also used in different ranges of applications such as rot mean square converter, an analog-to-digital converter, etc.

**What is Antilog?**

**Antilog Amplifier**

An Op-amp in which the output voltage of the amplifier (V_{0}) is directly proportionate to the anti-log of the input voltage (V_{i}) is known as an anti-logarithmic amplifier or anti-log amplifier. Here, we are going to discuss the operational amplifier configuration that forms the anti-logarithmic amplifier in detail.

**Antilog Amplifier Circuit**

**Antilog Amplifier Using Transistor**

**Antilog Amplifier using Diode**

In the antilog amplifier, the input signal is at the inverting pin of the operational amplifier, which passes through a diode.

**Output and Working Principle of Antilog Amplifier**

As observed in the circuit shown above, the negative feedback is achieved by connecting the output to the inverting input terminal. According to the concept of the virtual ground between the input terminals of an amplifier, the voltage V_{1} at the inverting terminal will be zero. Because of ideally infinite input impedance, the current flowing through the diode due to the applied input voltage in the inverting terminal will not enter the op-amp; instead, it will flow along the feedback path through the resistor R as shown in the figure.

The compliment or inverse function of the logarithmic amplifier is ‘exponential’, anti-logarithmic or simply known as ‘antilog’. Consider the circuit given in the figure. The diode current is

Where, V_{D} is the diode voltage. According to the concept of virtual ground, V_{1}=0 as the non-inverting terminal is grounded as shown in the figure. Therefore the voltage across the diode can be expressed as V_{D }= V_{i }– V_{1} or V_{D} = V_{i} Hence, the current through the diode is

Due to the ideal characteristics of an op-amp (infinite input impedance), the current flowing through the diode ( i_{D}) flows along the feedback path through the resistor R, as we can observe in the figure.

Therefore i_{D }= i_{2}

And, V_{0} = -i_{2}R = -i_{D}R

Replacing i_{D} in the above equation we get

The parameters n, V_{T} and I_{S }are constants (they are only depend on the diode characteristics which are always constant for a particular diode). Therefore if the value of the feedback resistor R is fixed, then the output voltage V_{0} is directly proportional to the natural anti-logarithm (exponential) of the applied input voltage V_{i}. The above equation then can be simply represented as

Where K = – I_{S}R and a =

Therefore we can notice that the anti-logarithmic op-amp produces its output signal as the exponential value of the input voltage signal applied.

The gain of the anti-log amplifier is given by the value of K that is equal to -I_{S}R.

The –ve sign point out that there is a phase difference of 180degrees between the applied input s and the output of the anti-log amplifier.

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