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Although it is not specifically stated that a mileage log be maintained, the following extract from the IRS rules does require that the vehicles use be determinable. To properly accomplish this, a mileage log should be maintained. This will substantiate the "no personal use" requirement of the agreement between the employer and the employee. It will also substantiate appropriate business use of this vehicle. Here is the extract from the IRS. It also states a specific value to be used for each commute rather than requiring fuel use as a part of the equation.
Under this rule, you determine the value of a vehicle you provide to an employee for commuting use by multiplying each one-way commute (that is, from home to work or from work to home) by $1.50. If more than one employee commutes in the vehicle, this value applies to each employee. This amount must be included in the employee's wages or reimbursed by the employee.
You can use the commuting rule if all the following requirements are met.
You provide the vehicle to an employee for use in your trade or business and, for bona fide noncompensatory business reasons, you require the employee to commute in the vehicle. You will be treated as if you had met this requirement if the vehicle is generally used each workday to carry at least three employees to and from work in an employer sponsored commuting pool.
You establish a written policy under which you do not allow the employee to use the vehicle for personal purposes other than for commuting or de minimis personal use (such as a stop for a personal errand on the way between a business delivery and the employee's home). Personal use of a vehicle is all use that is not for your trade or business.
The employee does not use the vehicle for personal purposes other than commuting and de minimis personal use.
If this vehicle is an automobile (any four-wheeled vehicle, such as a car, pickup truck, or van), the employee who uses it for commuting is not a control employee.