Now there is no specific place where the Bible explicitly says the earth is 6000 years old. If it did it would be 6000 years plus however many years since that was written. However, the Bible does give sufficient chronological data that you can calculate back to where the creation would be. When you do this, you get a figure of around 6000 years.
One of the common ways of trying to get around this is to claim that there was a gap between Genesis 1:1 and Genesis 1:2. However reading these verses shows no indication of such a gap.
1, In the beginning God created the heaven and the earth.
2, And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.
As you can see from reading this passage, one reason there is no evidence of a gap is because this is not a chronological description. Verse one is simply saying what God did and verse two is describing the initial conditions of the Earth. You cannot have a chronological gap, where there is no time difference being described.
One workaround is to claim that there are gaps in the genealogies found in Genesis 5 and 11. One of the excuses for this, is the fact that in Jesus’ genealogy in the gospel of Luke there is an extra Cainan in the genealogy in chapter 3, that is not present in Genesis 11. First of all, this admission may just suggest that Cainan died before Salah was born and so Salah was raised by his grandfather. In any case, you would have to have an awful lot of gaps, to add a significant amount of time. The genealogy would make no sense if it was just a small fraction of the actual list.
Another common argument is to claim that the days of Genesis are not literal 24-hour days. Those who make this argument can give no scriptural reason, for equating these days to millions of years, meanwhile the passage itself clearly refers to evening and morning. This approach takes two forms. The first of which is that the day simply equals long periods (day-age theory) of sequential time. The other claims that they refer to periods of creation (progressive creation) that can overlap resulting in no real chronological order,
The final workaround is simply a claim that Genesis 1 - 3 is just an allegory or myth. This view is generally held by theistic evolutionists. Most of those who hold these positions are also more liberal theologically, and often politically, while still claiming to be Christians. One interesting thing about this view is that you can never get a straight answer from those who hold this position just when the Bible starts becoming real history. Not only is there no agreement among these people on this question, but when asked most of them will not even try to answer. The reason for this is that they do not really have one, and at the same time they don't really care. Some of them go as far as to say that none of it is truly historical.
The point is that even though you do not find any place in the Bible where it says that on this particular day the earth was such and such years old, it does provide all the chronological data that you need to come within a few years of the exact age. Furthermore, when you look at the age of the Earth given by evolutionists, it is based on a totally naturalistic model of the origin of the earth, there is no place for God in this model, in fact, if God did create the earth as he said, the earth could be any age younger than the 4.5-billion-year figure. Simply put if the Biblical account is correct, then the age based on atheistic mythology is not correct. Furthermore, the Bible gives us the chronological information that we need to know, to see about when God created the heavens and the earth.