I'm not understanding the concept of L-Pad attenuator other than seeing it as another ordinary voltage divider to attenuate a voltage. Electronics-Tutorials: L-pad Attenuator gives the following description:

L-pad attenuators are commonly used in audio applications to reduce a larger or more powerful signal while matching the impedance between the source and load in provide maximum power transfer. However, if the impedance of the source is different to the impedance of the load, the L-pad attenuator can be made to match either impedance but not both.

I understand maximum power transfer occurs when input impedance matches output impedance (or \$ Z_{in} = Z_{out} \$), and that \$ \text{dB} = 20log({V_{out} \over V_{in}}) \$. In terms of impedance matching, looking into the attenuator from the left, \$ Z_{in} = (Z_L || R_2)+R_1 \$ and from the right, \$ Z_{out} = Z_s + R_1 + R_2 \$ in series. Hence for the two impedances to match and if I'm given a voltage attenuation ratio, I'll have exactly the following two equations to solve for two unknown values of \$ R_1 \$ and \$ R_2 \$:

1. Matching input and output impedance: \$ (Z_L || R_2)+R_1 = Z_s + R_1 + R_2 \$.
2. Voltage attenuation ratio(): \$ V_{out} = V_{in} \big({{R_2 || Z_L} \over {({R_2 || Z_L})+Z_S+R1}}\big) \$

Aren't these two equations all I need to match input and output impedance while achieving a specific attenuated voltage? Why are there these logarithmic equations with a 'K' value solving for \$ R_1 \$ and \$ R_2 \$? And what does it mean when they say

L-pad attenuator can be made to match either impedance but not both

I'm not understanding the concept of L-Pad attenuator other than seeing it as another ordinary voltage divider to attenuate a voltage. Electronics-Tutorials: L-pad Attenuator gives the following description:

L-pad attenuators are commonly used in audio applications to reduce a larger or more powerful signal while matching the impedance between the source and load in provide maximum power transfer. However, if the impedance of the source is different to the impedance of the load, the L-pad attenuator can be made to match either impedance but not both.

I understand maximum power transfer occurs when input impedance matches output impedance (or \$ Z_{in} = Z_{out} \$), and that \$ \text{dB} = 20log({V_{out} \over V_{in}}) \$. In terms of impedance matching, looking into the attenuator from the left, \$ Z_{in} = (Z_L || R_2)+R_1 \$ and from the right, \$ Z_{out} = (Z_s + R_1) || R_2 \$ in series. Hence for the two impedances to match and if I'm given a voltage attenuation ratio, I'll have exactly the following two equations to solve for two unknown values of \$ R_1 \$ and \$ R_2 \$:

1. Matching input and output impedance: \$ (Z_L || R_2)+R_1 = (Z_s + R_1) || R_2 \$.
2. Voltage attenuation ratio(): \$ V_{out} = V_{in} \big({{R_2 || Z_L} \over {({R_2 || Z_L})+Z_S+R1}}\big) \$

Aren't these two equations all I need to match input and output impedance while achieving a specific attenuated voltage? Why are there these logarithmic equations with a 'K' value solving for \$ R_1 \$ and \$ R_2 \$? And what does it mean when they say

L-pad attenuator can be made to match either impedance but not both

I'm not understanding the concept of L-Pad attenuator other than seeing it as another ordinary voltage divider to attenuate a voltage. Electronics-Tutorials: L-pad Attenuator gives the following description:

L-pad attenuators are commonly used in audio applications to reduce a larger or more powerful signal while matching the impedance between the source and load in provide maximum power transfer. However, if the impedance of the source is different to the impedance of the load, the L-pad attenuator can be made to match either impedance but not both.

I understand maximum power transfer occurs when input impedance matches output impedance (or \$ Z_{in} = Z_{out} \$), and that \$ \text{dB} = 20log({V_{out} \over V_{in}}) \$. In terms of impedance matching, looking into the attenuator from the left, \$ Z_{in} = (Z_L || R_2)+R_1 \$ and from the right, \$ Z_{out} = Z_s + R_1 + R_2 \$ in series. Hence for the two impedances to match and if I'm given a voltage attenuation ratio, I'll have exactly the following two equations to solve for two unknown values of \$ R_1 \$ and \$ R2 \$

1. Matching input and output impedance: \$ (Z_L || R_2)+R_1 = Z_s + R_1 + R_2 \$.
2. Voltage attenuation ratio(): \$ V_{out} = V_{in} \big({{R_2 || Z_L} \over {({R_2 || Z_L})+Z_S+R1}}\big) \$

Isn't that really all I need? What are these logarithmic equations with a 'K' value for \$ R_1 \$ and \$ R_2 \$ doing here? And what does it mean when they say

L-pad attenuator can be made to match either impedance but not both

I'm not understanding the concept of L-Pad attenuator other than seeing it as another ordinary voltage divider to attenuate a voltage. Electronics-Tutorials: L-pad Attenuator gives the following description:

L-pad attenuators are commonly used in audio applications to reduce a larger or more powerful signal while matching the impedance between the source and load in provide maximum power transfer. However, if the impedance of the source is different to the impedance of the load, the L-pad attenuator can be made to match either impedance but not both.

I understand maximum power transfer occurs when input impedance matches output impedance (or \$ Z_{in} = Z_{out} \$), and that \$ \text{dB} = 20log({V_{out} \over V_{in}}) \$. In terms of impedance matching, looking into the attenuator from the left, \$ Z_{in} = (Z_L || R_2)+R_1 \$ and from the right, \$ Z_{out} = Z_s + R_1 + R_2 \$ in series. Hence for the two impedances to match and if I'm given a voltage attenuation ratio, I'll have exactly the following two equations to solve for two unknown values of \$ R_1 \$ and \$ R_2 \$:

1. Matching input and output impedance: \$ (Z_L || R_2)+R_1 = Z_s + R_1 + R_2 \$.
2. Voltage attenuation ratio(): \$ V_{out} = V_{in} \big({{R_2 || Z_L} \over {({R_2 || Z_L})+Z_S+R1}}\big) \$

Aren't these two equations all I need to match input and output impedance while achieving a specific attenuated voltage? Why are there these logarithmic equations with a 'K' value solving for \$ R_1 \$ and \$ R_2 \$? And what does it mean when they say

L-pad attenuator can be made to match either impedance but not both