INTRODUCTION

I'll use this model to explain how the blend modes work. Along with X axis, base image is the value of the pixel from 0 to 1 (0 to 255 or 0% to 100%).

The base image will be blended with the active image that has the brightness value increasing from 0 to 1 gradually.

Below, you will see when the base images have been blended with active images at the different modes.

By understanding these graphs, you will know how each pixel value from the base image will be if you blend with the other image on top.

The dash line is the plotted value from the result after increase the active image value from 0 to 1.
R = result B = base image (lower image in AE) A = active image (upper image in AE)
1-B, 1-A = inverted B, inverted A

ADDR = B+AResult becomes brighter but white will be clipped — **ADD**
*R = B+A*
Result becomes brighter but white will be clipped

SUBTRACTR = B-AResult becomes darker but black will be clipped. — **SUBTRACT**
*R = B-A*
Result becomes darker but black will be clipped.

*NOTE Mathematically, addition and subtraction are the opposite functions as can see in those graphs above. But to manipulate the image, it doesn't work like you would think. While Add mode makes image brighter by increasing the value of active image. Subtract mode makes image darker if the active image become brighter too. Which contradicts our common sense that to make the image darker we should apply the darker value to the base image.

DARKENING MODES (0<=R<=B)

These modes will produce the result image in the range of darker or the same as base image.
Increase value on active image, result image will be the same as base image.
Decrease value on active image, result will be darker.

DARKENR = min(B,A)Image becomes darker by clips off the certain value dictated by the active image. For example, If active image pixel value  is 0.5. The result image's brightest point will be cut off at 0.5. — **DARKEN**
*R = min(B,A)*
Image becomes darker by clips off the certain value dictated by the active image. *For example, If active image pixel value is 0.5. The result image's brightest point will be cut off at 0.5.*

MULTIPLYR = BxADarker and lower the contrast (clamping the slope down) without clipping black value. For example, if active image is 0.5. Result image's brightest value will be clamped to 0.5. — **MULTIPLY**
*R = BxA*
Darker and lower the contrast (clamping the slope down) without clipping black value. *For example, if active image is 0.5. Result image's brightest value will be clamped to 0.5.*

COLOR BURNR = 1-[(1-B)÷A]Darkening by crushing the contrast (increasing slope). Black value dictated by the active image. For example, if active image is 0.5. At 0.5 of base image will be black while the base image's white point stay the s… — **COLOR BURN**
*R = 1-[(1-B)÷A]*
Darkening by crushing the contrast (increasing slope). Black value dictated by the active image. *For example, if active image is 0.5. At 0.5 of base image will be black while the base image's white point stay the same value.*

LINEAR BURNR = 1-[(1-B)+(1-A)]Result image becomes darker by lower the pixel value linearly and let the black value clipped. For example, if active image is 0.5. At 0.5 of base image will be darken to black and the white point will be reduced to 0.5. — **LINEAR BURN**
*R = 1-[(1-B)+(1-A)]*
Result image becomes darker by lower the pixel value linearly and let the black value clipped. *For example, if active image is 0.5. At 0.5 of base image will be darken to black and the white point will be reduced to 0.5.*

BRIGHTENING MODES (B<=R<=1)

These modes will produce the result image in the range of brighter or the same as base image.

LIGHTENR = max(B,A)Image becomes brighter by clips off the certain value dictated by the active image — **LIGHTEN**
*R = max(B,A)*
Image becomes brighter by clips off the certain value dictated by the active image

SCREENR = 1-[(1-B)x(1-A)]R = [(1-B)xA]+BBrighter and lower the contrast (clamping the slope down) without clipping white value. — **SCREEN**
R = 1-[(1-B)x(1-A)]
R = [(1-B)xA]+B
Brighter and lower the contrast (clamping the slope down) without clipping white value.

COLOR DODGER = B÷(1-A)Brightening by crushing the contrast (increasing slope). White value dictated by the active image. — **COLOR DODGE**
*R = B÷(1-A)*
Brightening by crushing the contrast (increasing slope). White value dictated by the active image.

LINEAR DODGER = B+AResult image becomes brighter by increase the pixel value linearly and let the white value clipped. This is Add mode, actually. — **LINEAR DODGE**
*R = B+A*
Result image becomes brighter by increase the pixel value linearly and let the white value clipped. This is Add mode, actually.

From here, you can see that Darkening and Brightening mode are pairs of each other, Darken - Lighten, Multiply - Screen, Color Burn - Color Dodge, Linear Burn - Linear Dodge.

DARKER TO BRIGHTER MODES (0<=R<=1)

OVERLAYB < 0.5 ; 2(BxA) B > 0.5 ; [(1-B)xA]+[B-(1-B)]At 0-0.5 of base layer pixel value, it will perform 2x Multiply mode to base layer. At 0.5-1 of base layer pixel value, it will perform Clamped Screen mode. — **OVERLAY**
B < 0.5 ; 2(BxA)
B > 0.5 ; 1-2{(1-B)x(1-A)}
At 0-0.5 of base layer pixel value, it will perform 2x Multiply mode to base layer. At 0.5-1 of base layer pixel value, it will perform Clamped Screen mode.

SOFT LIGHTLike overlay mode but have a smooth transition where the base pixel value moving closer to 0.5. So, it is not clipping black or white like overlay does. This creates softer S-curve than overlay. — **SOFT LIGHT**
(1-2b)a^2 + 2ba
Like overlay mode but has a smooth transition where the base pixel value moving closer to 0.5. So, it does not clipping black or white like overlay does. This creates softer S-curve than overlay.

PIN LIGHTA < 0.5 ; DarkenA > 0.5 ; Lighten — **PIN LIGHT**
A < 0.5 ; Darken
A > 0.5 ; Lighten

HARD LIGHTA < 0.5 ; MultiplyA > 0.5 ;Screen — **HARD LIGHT**
A < 0.5 ; Multiply
A > 0.5 ;Screen

VIVID LIGHTA < 0.5 ; Color burnA > 0.5 ; Color dodge — **VIVID LIGHT**
A < 0.5 ; Color burn
A > 0.5 ; Color dodge

LINEAR LIGHTA < 0.5 ; Linear burnA > 0.5 Linear dodge — **LINEAR LIGHT**
A < 0.5 ; Linear burn
A > 0.5 Linear dodge

UTILITIES MODES

HARD MIXB < 1-A ; R = 0B > 1-A ; R = 1Result will be 0 or 1 (255) only.- Create matte from z-depth pass. — **HARD MIX**
B < 1-A ; R = 0
B > 1-A ; R = 1
Result will be 0 or 1 (255) only.
**- Create matte from z-depth pass.**

DIFFERENCER = abs(A-B)Find any base pixel that has the same value to active pixel and turn into black. Any pixel in between will be move in linear fashion. - Compare image- Find edge — **DIFFERENCE**
*R = abs(A-B)*
Find any base pixel that has the same value to active pixel and turn into black. Any pixel in between will be move in linear fashion.
**- Compare image**
**- Find edge**

EXCLUSIONResult image looks similar to Difference but leave the mid tone untouched. — **EXCLUSION**
*Result image looks similar to Difference but leave the mid tone untouched.*

DIVIDER = B÷ARemove A out from B and turn into white- Color collect — **DIVIDE**
*R = B÷A*
Remove A out from B and turn into white
**- Color collect**

It is not necessary to remember all of these modes. But by knowing how they work, you can choose the right one or when something doesn't look right you know how to fix. Also, It prevents you from unnecessarily stacking all bunch of layers, gradient + color + levels + curves + again + again + + +. From my professional experience, I've seen these so many time.