The goal of transformations is to change the numeric scale that we use to represent the variable so that the values on the transformed scale more closely approximate the desired normal distribution or linear relationship with another variable.
Mathematically we can create a different but equivalent set of values for a numeric variable by performing the same arithmetic operation on each value in the set. For example, suppose we represent the earnings of three subjects on the metric variable INCOME as $10,000, $15,000, and $17,000. The interval between the INCOME of subject 1 and subject 2 is $5,000. The interval between INCOME for subject 2 and subject 3 is $2,000. If we add $1,000 to each subjects INCOME to create the transformed variable TINCOME, our three subjects have transformed values of $11,000, $16,000, and $18,000. The actual income of the three subjects did not change, but the way we represented their income on the TINCOME variable is different. The relationships between the three real income values are preserved in the transformed values, i.e. the interval between TINCOME of subject 1 and subject 2 is $5,000. The interval between TINCOME for subject 2 and subject 3 is $2,000.
We can reverse the arithmetic operation on the transformed variable to produce the value of the original variable. In fact, we need to be careful to distinguish between the measurement units of the original variable and the measurement units of the transformed variable. The transformed variable is used in the statistical calculations and the statistical output is expressed in transformed measurement units. In our interpretation, we must reverse the transformation when we want to discuss findings in the original measurement units.
For example, dependent variables like income and housing value are often expressed in logarithmic units because the distribution of the data is skewed by a few extremely large values. If we used our analysis to estimate the value of the dependent variable for some combination of independent variables, the estimate would be in logarithmic units. To make this a useful estimate, we would have to convert the logarithmic estimate back to a decimal value.