Tazz is explaining logarithms to Noah. Shall I give you an example?

In a nut shell: What power do I have to raise 2 in order to get 64? Or, 2

Which is bigger: 2

To solve, we will use the identity ln( a

Computing 2

How many digits are in 3

So 3

One last problem, which I've had to do: what is 3.14

In a nut shell: What power do I have to raise 2 in order to get 64? Or, 2

^{x}= 64, solve for x. In this question is the basis of a logarithms. The 2^{x}= 64 can also be expressed, x = log_{2}( 64 ) = 6, because 2^{6}= 64.Which is bigger: 2

^{64}or 10^{40}Classic pre-calc problem, and the problem Tazz is explaining to Noah.To solve, we will use the identity ln( a

^{x}) = x ln( a )ln( 2

ln( 10

ln( 2 ) ≈ 0.69

ln( 10 ) ≈ 2.3

64*0.69 = 44.16

40 * 2.3 = 92

44.16 < 92

Therefore, 2^{64}) = 64 * ln( 2 )ln( 10

^{40}) = 40 * ln( 10 )ln( 2 ) ≈ 0.69

ln( 10 ) ≈ 2.3

64*0.69 = 44.16

40 * 2.3 = 92

44.16 < 92

^{64}< 10^{40}Computing 2

^{64}isn't enough to overflow a calculator with 20 digits. But if you wanted to see if 3^{1000}is larger then 4^{700}then using a logarithm is a better option.How many digits are in 3

^{1000}? Again... let's use a logarithm. Log in base 10, or log_{10}, will tell us how many digits are in a number. Why? Think about this. 10^{3}= 10 * 10 * 10 = 1000--4 digits. 10^{8}= 100,000,000--9 digits. So digits = %u230Alog_{10}( n )%u230B + 1.10

%u230Ax%u230B + 1 = 478

^{x}= 3^{1000}log( 10^{x}) = log( 3^{1000})Use identity : log( a

x * log( 10 ) = 1000 * log( 3 )^{x}) = x log( a )log( 10 ) = 1

x = 1000 * log( 3 )log( 3 ) ≈ 0.4771

x = ( 1000 * 0.4771) = 477.13%u230Ax%u230B + 1 = 478

^{}

^{1000}has 478 digits. A more general formula is for the digits in a^{x}is d = %u230Aa * log_{10}( x )%u230B + 1One last problem, which I've had to do: what is 3.14

^{2.71}? .x = 3.14

^{2.71}ln( x ) = ln( 3.14

^{2.71 })Use identity : ln( a

^{x}) = x ln( a )ln( x ) = 2.71 * ln( 3.14 )

x = e

x = e

^{2.71 * ln( 3.14 )}ln( 3.14 ) = 1.442

x = e

If you are clever, you may have asked "That's great, but how do you calculate ln and e?". Turns out that this isn't all that hard, but that's for an other article. ^{2.71 * 1.442}= 22.22