How to calculate the input current of a logarithmic amplifier

The formula for computing the output voltage of a logarithmic amplifier such as the ADL5303 is the following

$$V_{\text{out}} = V_Y \cdot \log_{10}\left(\frac{I_{PD}}{I_{Z}}\right)$$

where:

• $V_{\text{out}}$ is the output voltage of the logarithmic amplifier
• $V_Y$ is the gain of the logarithmic amplifier, expressed in Volts per Decade.
• $I_{PD}$ is the input current of the logarithmic amplifier (PD stands for Photodiode)
• $I_{Z}$ is the reference current, also called intercept point of the logarithmic amplifier (Z stands for Zero). This is the current that would result in an output voltage of 0V.

Solving the equation for $I_{PD}$

We can solve this equation using the following steps:

Divide both sides by $V_Y$:

$$\frac{\text{V}_{\text{out}}}{V_Y} = \log_{10}\left(\frac{I_{PD}}{I_{Z}}\right)$$

Raise both sides as powers of 10:

$$10^{\frac{\text{V}_{\text{out}}}{V_Y}} = \frac{I_{PD}}{I_{Z}}$$

Multiply both sides by $I_{Z}$ to obtain the formula for output current:

$$I_{PD} = I_{Z} \cdot 10^{\frac{\text{V}_{\text{out}}}{V_Y}}$$