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# Log MAR: definition and interest

### Log MAR: definition

The LogMAR (or Log MAR) is a unit that allows to quantify theVisual acuity in order to perform calculations statistics such as average, deviation of the Visual acuity in a population of eyes, etc.

### Reminder: MAR

The MAR (minimum angle of resolution) is the unit that defines the separator (for a resolution) power of the eye. It corresponds to the minimum angle that can deal with two light sources with regard to the eye by being 'just' separate. S1 and S2 are their respective retinal images to separate light sources, and I1 and I2: depending on the angle α they form with the eye (N being the nodal point), the images are more or less separated. The minimum angle of resolution (or separation) is the one "in deca" from which the images of sources, represented in the form of "disks" light whose intensity schematic profile is also represented, are more separated. When sources form an angle equal to the MAR with the eye, they are 'just separate', and priori perceived as distinct if these 'geometrically' separate images are properly sampled.

His progression is geometric, and inversely proportional to that of Visual acuity (when the angle double MAR, acuity is divided by 2). Indeed, remember the decimal acuity (expressed in tenths) is in fact equal to the inverse of the Mar. For example, for a 1 min MAR, Visual acuity is 1 or 10/10. For a MAR of 4 minutes, Visual acuity is 1/4 What done in tenths: 2.5/10.

However the gap between two lines of Visual acuity on Snellen chart is not the same: there are more "gain" of resolution between 1 and 4 tenths (the angle of minimum resolution = MAR is divided by 4) that between 5 to 10 tenths)MAR is divided by 2).

### Progression of Visual acuity

The progression between two levels of acuity must be the same, regardless of the line which we "leave", and "get"! It is the role of the conversions in log MAR: a same gain of lines of Visual acuity (this gain is the reflection of a certain percentage of reduction of the MAR) is converted to a constant number of logMAR, regardless of the value of the MAR initial.

Indeed, as Visual acuity is the opposite of the angle MAR (10/10 = 1 is a minute of arc, 9/10 is 1.11 minute arc, 8/10 is 1.25 minute arc, 7/10 is 1.42 minute of arc,..., 2/10 = 0.5 is 5 minutes of arc, etc.) we note that between 2 lines of decimal acuity, reduction of Visual acuity is not constant. More progress from tenth in tenth with a classic Visual acuity of decimal progression scale, and the gain between each line is reduced. Here are the comparative gain between tenths of Visual acuity and reduction of the angle line MAR online:

1/10: MAR = 10 minutes

2/10: MAR = 5 minutes (half or 50% of the MAR angle reduction)

3/10: MAR = 3.33 minutes (33% of the MAR angle reduction)

4/10: MAR = 2.5 minutes (25% of the MAR angle reduction)

5/10: MAR = 2mn (reduction of 20% of the MAR angle)

6/10: MAR = 1.66 minutes (reduction of 16.7% of the MAR angle)

7/10: MAR = 1.42 minutes (reduction of 14.4% angle MAR)

8/10: MAR = 1.25 minutes (12% of the MAR angle reduction)

9/10: MAR = 1.11 minutes (reduction of 11.1% of the MAR angle)

10/10: MAR = 1 min (10% of the MAR angle reduction)

Difficult to make a calculation of average arithmetic with the tenth in a reference population. The use of the logarithm however lets find an arithmetic progression, at the cost of a change of variable.
Thus, thanks to the passage in log MAR, we obtain the same gain for an eye whoseVisual acuity increases a MAR 4 minutes of angle (either an acuity of 0.25 or 2.5/10) and for an eye growing acuity of a MAR of 2 minutes (either a 0.50 AV or 5/10) in a 1 min (1 or 10/10 AV) MAR a MAR of 8 minutes of angle or a sharpness of 0.13 in decimal (i.e. 1.3/10).  This is expected, since in these two case, Visual acuity has doubled (the value of the MAR has been divided by 2).

How to convert the MAR in logMAR?

LogMAR unit is defined as the logarithm in base 10 decimal Visual acuity.

-Log (10)MAR).

We back the decimal fraction of the sharpness to a power of 10. The MAR log is equal to the exponent (which we reverse the sign).

Two case are particularly easy to calculate:

* 10/10 = 1 = 100

-Log = 0 MAR

* 1/10 = 0.1 = 10-1

-Log = 1 MAR

* 5/10

It is less easy to mentally calculate the logMAR of 0.5 (5/10) (how much power should I raise 10 to get the value 0.5?)

5/10 = 0.5 = 10-0.3

-MAR = 0.3 log

Thanks to the properties of logarithms, between each 0.1 difference logMAR, the percentage of reduction or increase the angle of MAR is the same! In fact, 10-0.1 x 10-0.1 =10-0.2  and 10-0.3 x 10-0.1 =10-0.4 (exhibitors are added together). Between each line, for an addition of-0.1 LogMAR, MAR angle increases by 25% (multiply by 1.25).

In case of acuity greater than less than 1 minutes 10/10 (Tue), decimal acuity turnover is greater than 1, and the exponent of 10 greater than 0; (ex: LogMAR = + 0.3 for acuity of 20/10, is a MAR to 05 minutes of arc). As we reverse its sign, the end result is negative (-0.3).

Some scales of Visual acuity we were designed so that the gain between each line is the reflection of a same progression in terms of MAR. The EDTRS scale is the best known.

The following table allows to convert between the different Visual acuity rating:

### The Visual acuities conversion table

 Snellen notation MAR -logMAR Decimal 6/60 20/200 10 1.0 0.10 6/48 20/160 8.0 0.9 0.13 6/38 20/125 6.3 0.8 0.16 6/30 20/100 5.0 0.7 0.20 6/24 20/80 4.0 0.6 0.25 6/19 20/60 3.2 0.5 0.32 6/15 20/50 2.5 0.4 0.40 6/12 20/40 2.0 0.3 0.50 6/9.5 20/30 1.6 0.2 0.63 6/7.5 20/25 1.25 0.1 0.80 6/6 20/20 1.00 0.0 1.00 6/4.8 20/16 0.80 -0.1 1.25 6/3.8 20/12.5 0.63 -0.2 1.58 6/3.0 20/10 0.50 -0.3 2.00