Age of the Earth
The modern consensus is that the Earth is 4.543 billion years old. We have at least two good reasons for believing this. In the first place, this age forms the backbone of a geological model that is consistent with many diverse fields of study. And so far no other estimate comes close to explaining all the observations.
But more importantly, since the mid to late 20th century, there is a growing avalanche of unambiguous direct physical evidence.  In particular we know the composition of the Earth and that its internal heat is maintained by radioactive decay. And we can use this radioactivity to date the formation of rocks from liquid magma.
We also have rocks from the moon and meteorites from space with the same radioactive elements which also allows us to date their formation.
CONVERGENCE IN SCIENCE
The methodology of science is that after many observations, we create mathematical models which are effectively curve fits to tabular data. The challenge has been to resolve inconsistencies especially across disciplines originally thought unrelated. But over time understandings invariably converge to form a coherent whole. 
In one of the earliest musings, the first century Roman philosopher Lucretius, in his treatise “The Nature of Things” noted the earliest Greek writings dated only as far back as the Trojan War. With little allowance for pre-literate societies, he thus surmised creation occurred about 1000 B.C. Talmudic Jewish rabbis as well as such Protestant notables as Martin Luther, John Calvin and others improved on this estimate by relying on the Old Testament. They correctly noted the first five books of the Bible (1450-1290 B.C.)  were among the world’s oldest written records preserving oral traditions dating as far back as 2000 B.C. Summing the lineages gave a rough time span of about 6000 years. Somewhat whimsically, the Protestant Calvinist Archbishop James Ussher in 1684 calculated the definitive moment of creation as having occurred at 6 PM on the evening of October 22, 4004 B.C. 
We now know from deposition rates of sedimentary basins on Earth, from sea floor spreading and continental drift, from the nuclear physics of stellar evolution, from the distribution of craters on the moon and around the solar system, from the time animal breeders require to develop new characteristics providing a time scale for the natural evolution, and from many other observations, that the earth must have existed for at least hundreds of millions of years. But this truly vast expanse of time has only been accurately quantified within the last several centuries.
The best physical evidence for an ancient earth comes from the decay of radioactive elements in rocks. Radioactivity was discovered in 1896 by Henri Becquerel and the first rocks haphazardly dated by Rutherford and Beltwood by about 1905. But the modern value, which added billions of years to earlier estimates, was only developed in the 1950s and within the bounds of living memory. 
But paradoxically, rocks on earth are not useful for determining its precise age because, while ancient, their ages vary widely and nary a pebble appears older than about 3.8 billion years. This is because mountain ranges are born in volcanoes and then weathered away to form sedimentary basins. Later these may be buried under many layers of deposits which under heat and pressure slowly transform into metamorphic minerals. Eventually everything is recycled as plate tectonics subducts continents back into the mantle to be re-melted. And this process has apparently gone through many such cycles and continues today.
A very few rocks, however, contain zircon crystals which can survive being eroded out of their original matrix and incorporated into another rock later. Several tiny inclusions from Jack Hills, Australia appear to be older than 4 billion years. And baring as yet undetected contamination from a variety of sources, three of these tiny crystals have the world’s record of 4.4 billion years. 
Fortunately we also have a few well measured meteorites, about 70 or so but certainly less than 100, which display a consistent age of 4.55 to 4.57 billion years. Some of the more famous meteorites are the Canyon Diablo fall and the more recent Allende meteorite. The thinking is that while rocks on earth are constantly being created and recycled, rocks in space are unchanged since the very beginning of the solar system.
These dates are also consistent with the hundreds of pounds of moon rocks returned by the manned Apollo missions. Most of these seem to have been “re-worked” by a “Late Heavy Bombardment” lasting from about 4.1 to 3.8 billion years ago. During this interval the inner solar system, which had formed hundreds of millions of years earlier, experienced a large increase in the number of asteroid impacts significantly reworking their surfaces. Apparently this was caused by the tidal consequences of an orbital resonance between Jupiter and Saturn as they migrated outward from the sun. Fortuitously the consequent scattering of debris from the asteroid belt removed many large objects dangerous to the evolution of life on Earth. But again a few zircon crystals in the rocks from the moon date its formation to sometime before about 4.51 billion years ago.
To understand these processes, we need to first understand the radioactive decay process, the nature of isotopes, and the properties of atomic structures in rocks. These are described below.
When a radioactive element is extracted from its ore, it is observed to decay in proportion to the amount present. That is, if we double the amount of the sample, we get twice as many decay events per second. This is a consequence of quantum mechanics which states that identical atoms, and with everything we can measure about them being exactly the same, will nevertheless split into lighter elements spontaneously and unpredictably. The only thing we can say for certain is that each atom has an identical probability of a random decay event. This is expressed mathematically by the equation
where N(t) is the number of atoms at time t, the derivative operator d/dt gives the number of decay events per unit time, and λ is the decay rate constant. By integrating both sides we get the well known equation
For a surprisingly small sample, we can get an accurate average decay rate λ even when the half life is many billions of years. 
Also as its name suggests, the “half-life” is that time after which half of the original sample has decayed. We can compute this time, thalf, as follows
A few useful radioactive decay processes for dating the earth include the following
1. Samerium-147 decays to Neodymium-143 with a half life of 106 billion years. This has a stable isotope Neodymium-144 for comparison.
2. Rubidium-87 decays to Strontium-87 with a half life of 49.23 billion years. This has a stable isotope Strontium-86 for comparison.
3. Rhenium-187 decays to Osmium-187 with a half life of 43.0 billion years.
4. Lutetium-176 decays to Hafnium-176 with a half life of 35.9 billion years.
5. Thorium-232 decays to Lead 208 with a half life of 14.0 billion years
6. Uranium-238 decays to Lead-206 with a half life of 4.468 billion years. This has a stable isotope Lead-204 for comparison.
7. Potassium-40 decays to Argon-40 with a half life of 1.251 billon years. Note that Potassium-40 also decays to calcium.
8. Uranium-235 decays to Lead-207 with a half life of 0.704 billion years.
All atoms of a particular element have the same number of electrons which determines how they bond to other atoms. This gives them identical chemical properties. But surprisingly, atoms of the same element can have different masses or weights. We now realize this is because the number of neutrons in the atomic nucleus is different. And this makes sense because neutrons carry no electrical charge and have little influence on the circling electrons. These chemically identical atoms, but with different masses or weights are called “isotopes”.
The number of neutrons is however crucial to the long term stability of the atom. If there are too many or too few in the nucleus, the atom will be unstable sometimes with a very short radioactive half life. Fortunately with just the right number of neutrons, isotopes appear to be stable and as best we can determine will last forever.
THE NATURE OF ROCKS
When an igneous rock solidifies from molten lava, it crystallizes into a jumble of interlocking grains. Each grain is a tiny crystal meaning its atoms form straight and unbroken rows. Within these crystals the same arrangement of different elements is uniformly repeated. As a consequence elemental concentrations do not vary across the grain. But because of miniscule variations in the original melt, adjacent grains will have slightly different atomic structures and slightly different elemental concentrations.
But this is not the whole story because each element consists of different isotopes which are chemically indistinguishable. As a consequence, isotopes will be well mixed throughout the original melt before the rock formed and immediately after the rock solidified. So for instance if an element consists of two isotopes, one heavier and one lighter, and each isotope amounted to one half of that element in one grain, then this ratio would have been exactly the same for all grains.
So even though a total elemental concentration, while constant within a grain, varied between grains, the isotopic ratios would have been the same for all grains. And finally if one isotope is involved in radioactive decay and the other is not, then their ratio will gradually change over time. We are confident of this because of what we observe in rocks formed today.
RADIOACTIVE DATING WITH RUBIDIUM->STRONTIUM
Rubidium and strontium have many isotopes but several are well suited for radioactive dating. Rubidium-87, for instance, transmutes to Strontium-87 with a half life of about 49.23 billion years. On the other hand the isotopes Strontium-87 as well as Strontium-86 are stable. Note that “86” and “87” are the “mass numbers” which is the sum of protons plus neutrons in the nucleus.
To reiterate, Rubidium-87 decays into Strontium-87 when a neutron in its nucleus transmutes to a proton by emitting a “beta” particle (electron) or
So the concentration of Rubidium-87 decreases with time, t, as
And an equal amount of Strontium-87 is created so its concentration increases with time in direct proportion as
The immediate difficulty is that while we can measure the concentrations Sr87(t) and Rb87(t) today, we don’t know how much Sr87(0) was there when the rock solidified at time t=0. We have three variables and only one equation. To get around this problem, we recall that the isotope Strontium-86 is not radioactive. Since the concentration of Sr86 never changes we can write the trivial relation as
We can use this feature to break the logjam of too little information to compute the age. For any particular grain, we can divide by Sr86 which is independent of time as follows
The beauty of this equation is that “b” which is the isotopic ratio of “Sr87(0) / Sr86” is a constant for all grains because isotopes of the same chemical element must have been well mixed when the rock solidified. The result is an equation of a straight line where “m” is the constant slope at a given measurement time “t” and “b” is the constant y-axis intercept for all grains. The ratios of “Sr87(t) / Sr86” and “Rb87(t) / Sr86” which we measure today are slightly different in each grain but must lie along the straight line derived above.
By plotting measurements of many adjacent grains in the same rock at time “t” today, we get a straight line whose slope “m” is a measure of how long ago the rock solidified. And this slope increases for any given rock as time progresses. This is just as we expected and the straightness of these lines serves to validate the assumptions. Each line is called an “isochron” meaning all its points are measured after the same length of time.
The time “t” since the rock solidified is related to the slope “m” by the equation
or finally, the age of the rock is
To use radioactivity in the dating of rocks we have made the following important assumptions.
1. The ratio of isotopic concentrations for the same element is uniform across the entire rock when it was formed at time t = 0. This is verified by the straight line on the “Isochron” graph.
2. Nucleation of different minerals is highly sensitive to minute variations in temperature and atomic abundances so that each grain forms an ordered lattice with slightly different crystal structures and atomic abundances. This is well verified by x-ray diffraction studies.
3. Neither the original elements nor radioactive decay products leave the rock once it is formed. And the rock was not heated since its formation since this can jumble concentrations. Fortunately this is fairly easy to determine from a microscopic observation.
4. The radioactive decay rate of unstable isotopes has remained unchanged over billions of years.
The bedrock of Judeo-Christian theology has always been that the Bible was inspired but not dictated. Rather its sweeping visions sometimes use allegory and parable to teach timeless moral truths. And this is especially true of Genesis which is the first book of the Bible. Genesis alludes to the logical necessity of a creation event ex nihilo which by definition cannot have a physical basis. This is because it is scientifically impossible for a universe to create itself. And thus in reasoning consistent with Occam’s razor posits the existence of a Creator. Genesis further notes the unique nature of man’s consciousness and free will and moral sense, which to the extent they exist, must logically be indeterministic, or independent of strict natural law. That is there must be a supernatural component where “supernatural” simply means “above, or beyond, nature”.
And lest anyone doubt, these musing are in full agreement with modern science as we ever more accurately understand both its power and limits. As faith informs science so does science inform faith. For as St. Augustine, born in 354 A.D., noted in his commentaries, Genesis is a polemic against nature worship and especially superstition. Rather, in a unique vision, the Bible describes a natural world created with consistency and order, and which is amenable to human understanding. This was the very beginning of science and the first scientific revolution. 
Fundamentalists ignore the rational basis of Christianity or “logos” from John’s testament. They ignore St. Augustine’s admonition that there is only one truth and that one’s interpretation of sacred scripture can never be in conflict with reason or physical observation. They reject the conventional Christian concept of “secondary causation” as maintained by the Catholic Church which was a necessary precursor to science. Most importantly, fundamentalists demand the Bible be taken on strictly literal grounds as a practical treatise on the shipbuilding of arks and the biology of whales and the formation of the universe and the solar system, hopelessly diluting the moral message.
On the other hand, the so-called scientific atheists continue to deny the tenants of modern science. Some of these rejected principles include Gödel’s theorems as a consequence of the axiomatic foundation of mathematics, the limits of logic from Bertrand Russell and others, entropy, chaos theory, quantum mechanics, evidence for the Big Bang, the logical impossibility of absolute knowledge or a “Theory of Everything”, the very nature of the scientific method, and so forth. 
Amplifying this silliness but in a funny convergence, fundamentalists and atheists both believe a definitive test of the Christian message is whether or not the world is 6000 years old. Considering this ignorance of mainstream religion and science by both groups on both subjects, it is hard to take either seriously.
1. Images courtesy of “Earthrise, NASA Image AS8-14-2383 taken by Astronaut W. Anders on Apollo 8 and “NASA Layered Earth Image” from https://upload.wikimedia.org/wikipedia/commons/1/1b/Earth_layers_NASA.png
2. New scientific theories make assumptions about some underlying order in the natural world and then derive mathematical equations that are necessary consequences. To the extent these equations predict future behavior, especially in unrelated matters, we have confidence in the theory. When discrepancies arise, we eventually find new assumptions leading to new equations but most importantly to a new scientific world view as well. So scientific advances are not so much a gradual improvement in understanding as much as a chaotic lurching from curve fit to new curve fit.
A case in point was Newton’s gravitational forces acting at a distance, invented to describe the motions of the heavens and cannonballs on earth. When these failed to predict the location of the planet Mercury by a truly trivial amount each century, we completely discarded the “clockwork universe” in favor of Einstein’s General Relativity. Gravitational fields are now believed not to exist at all. Rather we believe in a space-time warped in an unseen dimension. Except of course that quantum mechanics and General Relativity are in conflict, so at least one of these must be somehow wrong as well.
3. Writing gradually evolved in Sumeria from 3500 to 3000 B.C. One of the oldest literary works is the Epic of Gilgamash, King of Uruk of the Third Dynasty of Ur, around 2100 B.C. The code of Hammurabi dates back to perhaps 1780 B.C. But despite a smattering of fragmentary works, by far the most extensive ancient writing is the Bible being first composed either in 1450 B.C. or 1290 B.C.
4. Jackson, P.W., “The Chronologers´Quest. The Search of the Age of the Earth”, 2006, Cambridge Press, page 291.
5. Hay, William, “Experimenting on a Small Planet: A Scholarly Entertainment”, 2012, Springer, page 63.
6. Patterson, Claire C., “Age of meteorites and the earth”, Geochimica et Cosmochimica Acta, 1956, Vol 10, pp. 230-237, Pergamon Press Ltd., London. Xx
7. McSween, Harry Y., “Meteorites and Their Parent Planets”, 1987, Cambridge University Press.
8. Valley, John W. et al., “Hadean age for a post-magma-ocean zircon confirmed by atom-probe tomography”, Nature Geoscience, Vol. 7, pages 219-223.
9. The measurement of radioactive half-lives is easily accomplished even on scales of longer than billions of years. This is because there are such a large number of atoms in even a modest amount of material, that we get a statistically significant number of decay events each second. For instance, for Rubidium-87 with a half life of tens of billions of years, we can calculate
Density of Rubidium-87 D = 1.532 g/cm3
Atomic Mass of Rubidium-87 M = 86.909 g/mole
Avogadro’s Number NA = 6.0221 * 1023 atoms/mole
Number of Atoms of Rubidium-87 NRb = NA * (D/M) = 1.0612 * 1022 atoms/cm3
For illustration we will now run the calculations backward. That is we will use the previously calculated half life to back out the original measurement of decay events per second for a given amount of material. These numbers are
Half life of Rubidium-87 thalf = 49.23 billion years
Number of seconds per year nsec = 31.557 million seconds/year
So finally we can calculate the number of radioactive decay events per second for a cubic centimeter of Rubidium-87 as
Number of Events d (Nevent) /dt = ln(2) * NRb / (thalf * nsec) = 4734 events / (second * cm3)
which is large enough to be accurately measured in a laboratory.
10. The ancient Greeks noted that while things decayed on earth, food rotted, floods swept away settlements, people aged and died, the heavens were never changing and apparently perfect. Accordingly they did not believe nature was either orderly or consistent or even that natural law existed at all. Rather Aristotle taught that only ideas reflecting “perfect forms” were worthy of study. This led to significant advances in logic and mathematics but was aggressively antagonistic towards science. The earlier work of Genesis on the other hand dramatically and diametrically opposed this world view.
11. The difficulty for atheists is that each and every one of the scientific disciplines which they reject, generally reinforce the Biblical world view of Genesis. So as our understanding of nature increases, atheistic-science as an explanation for everything must ever more retreat into the gaps of the unknown. This is perhaps why every new whacko proposal, to include cyclical universes, multi-worlds, many-worlds, determinism in quantum mechanics, quantum singularities out of nothing, and whatnot are so uproariously celebrated in the atheist press. Unfortunately each and every one of these crackpot musings, which attempt to dispel the Biblical world view, also fail to predict what science can already predict. That they are so obviously in violation of the vast majority of scientific theories and make predictions so grossly at odds with most scientific observations is perhaps the final strike against them. That a few declare that none of their predictions are theoretically observable, is more reminiscent of pagan superstition than rational thought, and is hardly a convincing recommendation either.