Can someone explain how 2(5)^14 ≈ 1.22(10^10) ?

2 × 5^14
= 2 × 5^4 × 5^10
= 2 × 5^4 × 5^10 × 2^10 / 2^10
= (2 × 5^4 / 2^10) × (5^10 × 2^10)
= (5^4 / 2^9) × 10^10
≈ 1.22 × 10^10

Another way:

2*(5 ^ 14) = 2* 5^4 * 5^10 = 1250 * 5^10
= (1250/1024) * (1024) * 5^10
= (1250/1024) * 2^10 * 5^10
= (1250/1024) * 10^10
= about 1.22 * 10^10

x = 2 * (5)^(14)
Calculate the logarithm base 10 of both sides of the equation:

log(x) = log(2) + 14 * log(5)

log(x) =~ 10 + 0.08661006
Cancel the logarithm base 10 to isolate "x" which is your original value like so:

10^[log(x)] = 10^[10 + 0.08661006]

x =10^(0.08661006) * 10^(10)

=~ 1.22 * 10^(10)

I suppose this could be helpful if your calculator can calculate logarithms but overflows with numbers greater than 10 billion.
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Pretty sure this works pretty well for very large numbers.

y = 2 * 5^(184)

log(y) = log(2) + 184 * log(5)

log(y) =~ 128.9115108
Pull out fractional and whole parts:

log(y) = 128 + 0.9115108

10^[log(y)] = 10^(0.9115108) * 10^(128)

y = 8.156 * 10^(128)

The Google calculator says this is about right so it must be right. My calculator overflows.

2(5)^14
= 2 × (5)^4 × (5)^10
= 1250 × 5^10
= 625/512 × 10^10
= 1.220703125 × 10^10

Getting an answer with three significant figures is a very good "approximation". So good in fact, I would just do the exact computation on a calculator.
2 * 5^14 = 1.220703125 * 10^10

F = 2(5)^14. (2^13)F = 10^14. F = (10^14)/(2^13) = (10^10)*[(10^4)/(2^13)]. Now 2^13 =
(2^3)(2^10) = 8(1024) = 8192. Then F = [(10^4)/8192](10^10) = [1.220703125](10^10). So
F is approximately = 1.22(10^10) to 3 sig. fig.

2 * 5^(14) = x * 10^(y)

5^(14) =>
(2 * 5)^(14) / 2^(14) =>
10^(14) / 2^(14)

2 * 5^(14) =>
2 * 10^(14) / 2^(14) =>
2^(-13) * 10^(14)

2^(-13) = 2^(-10) * 2^(-3) = (1/8) * (1/1024) = 1 / 8192

10^(14) / 8192 =>
10^(4) * 10^(10) / 8192 =>
(10000 / 8192) * 10^(10) =>
(2^4 * 5^4 / 2^13) * 10^(10) =>
(5^4 / 2^9) * 10^(10)

5^4 = 625
2^9 = 512

625 / 512 =>
312.5 / 256 =>
156.25 / 128 =>
78.125 / 64 =>
39.0625 / 32 =>
19.53125 / 16 =>
9.765625 / 8 =>
4.8828125 / 4 =>
1.220703125

1.220703125 * 10^(10)
or
1.22 * 10^(10)