To solve this logarithmic equation, we need to use the properties of logarithms.

Given that lg2 = a and lg3 = b, we want to find log(12)48 which means log base 12 of 48.

First, we can express 48 as a product of 2 and 24: 48 = 2 24. Using the log product rule, we can express log(12)48 as log(12)(2 24) = log(12)2 + log(12)24.

Since we know that lg2 = a, we can substitute a into log(12)2 to get a. Similarly, since lg3 = b, we can substitute b into log(12)3 to get b.

Therefore, log(12)48 = a + blog(12)24.

Now, we need to express log(12)24 in terms of a and b. Since 24 = 3 8 and log(12)3 = b, and log(12)8 can be expressed as log(12)(2 2 2), we can use the log product rule to find log(12)24 = log(12)(2 2 2 3) = log(12)(2 2 2) + log(12)3 = 3log(12)2 + b = 3a + b.

Substitute log(12)24 = 3a + b back into log(12)48 = a + blog(12)24 to get log(12)48 = a + b(3a + b) = a + 3ab + b^2.

Therefore, the final answer of log(12)48 in terms of a and b is a + 3ab + b^2.