# Log And Anti Log Amplifiers

The electronic circuits which perform the mathematical operations such as logarithm and anti-logarithm (exponential) with an amplification are called as **Logarithmic amplifier** and **Anti-Logarithmic amplifier** respectively.

This chapter discusses about the **Logarithmic amplifier** and **Anti-Logarithmic amplifier** in detail. Please note that these amplifiers fall under non-linear applications.

## Logarithmic Amplifier

A **logarithmic amplifier**, or a **log amplifier**, is an electronic circuit that produces an output that is proportional to the logarithm of the applied input. This section discusses about the op-amp based logarithmic amplifier in detail.

An op-amp based logarithmic amplifier produces a voltage at the output, which is proportional to the logarithm of the voltage applied to the resistor connected to its inverting terminal. The **circuit diagram** of an op-amp based logarithmic amplifier is shown in the following figure −

In the above circuit, the non-inverting input terminal of the op-amp is connected to ground. That means zero volts is applied at the non-inverting input terminal of the op-amp.

According to the **virtual short concept**, the voltage at the inverting input terminal of an op-amp will be equal to the voltage at its non-inverting input terminal. So, the voltage at the inverting input terminal will be zero volts.

The **nodal equation** at the inverting input terminal’s node is −

The following is the **equation for current** flowing through a diode, when it is in forward bias −

where,

IsIs is the saturation current of the diode,

VfVf is the voltage drop across diode, when it is in forward bias,

VTVT is the diode’s thermal equivalent voltage.

The **KVL equation** around the feedback loop of the op amp will be −

Substituting the value of VfVf in Equation 2, we get −

Observe that the left hand side terms of both equation 1 and equation 3 are same. Hence, equate the right hand side term of those two equations as shown below −

Applying **natural logarithm** on both sides, we get −

Note that in the above equation, the parameters n, VTVT and IsIs are constants. So, the output voltage V0V0 will be proportional to the **natural logarithm** of the input voltage ViVi for a fixed value of resistance R1R1.

Therefore, the op-amp based logarithmic amplifier circuit discussed above will produce an output, which is proportional to the natural logarithm of the input voltage VTVT, when R1Is=1VR1Is=1V.

Observe that the output voltage V0V0 has a **negative sign**, which indicates that there exists a 180^{0} phase difference between the input and the output.

## Anti-Logarithmic Amplifier

An **anti-logarithmic amplifier**, or an **anti-log amplifier**, is an electronic circuit that produces an output that is proportional to the anti-logarithm of the applied input. This section discusses about the op-amp based anti-logarithmic amplifier in detail.

An op-amp based anti-logarithmic amplifier produces a voltage at the output, which is proportional to the anti-logarithm of the voltage that is applied to the diode connected to its inverting terminal.

The **circuit diagram** of an op-amp based anti-logarithmic amplifier is shown in the following figure −

In the circuit shown above, the non-inverting input terminal of the op-amp is connected to ground. It means zero volts is applied to its non-inverting input terminal.

According to the **virtual short concept**, the voltage at the inverting input terminal of op-amp will be equal to the voltage present at its non-inverting input terminal. So, the voltage at its inverting input terminal will be zero volts.

The **nodal equation** at the inverting input terminal’s node is −

We know that the equation for the current flowing through a diode, when it is in forward bias, is as given below −

Substituting the value of IfIf in Equation 4, we get

The KVL equation at the input side of the inverting terminal of the op amp will be

Substituting, the value of in the Equation 5, we get −

Note that, in the above equation the parameters n, VTVT and IsIs are constants. So, the output voltage V0V0 will be proportional to the **anti-natural logarithm** (exponential) of the input voltage ViVi, for a fixed value of feedback resistance RfRf.

Therefore, the op-amp based anti-logarithmic amplifier circuit discussed above will produce an output, which is proportional to the anti-natural logarithm (exponential) of the input voltage ViVi when, RfIs=1VRfIs=1V. Observe that the output voltage V0V0 is having a **negative sign**, which indicates that there exists a 180^{0} phase difference between the input and the output.