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3 years ago

How does adding the log together automatically mean that it is a factorial?

Mathematics

1 answer:

andreyandreev [35.5K]3 years ago

6 0

Answer: The answer is given below.

Step-by-step explanation: We are given an equality involving logarithm and we are to show the implication of L.H.S. to R.H.S.

We will be using the following two properties of logarithm:

$(i)~\log_ba=\dfrac{1}{\log_ab},\\\\\\(ii)~log_ab+\log_ac=\log_a(bc).$

The proof is as follows:

$L.H.S.\\\\\\=\dfrac{1}{\log_2N}+\dfrac{1}{\log_3N}+\dfrac{1}{\log_4N}+\cdots+\dfrac{1}{\log_{100}N}\\\\\\=\log_N2+\logN3+\log_N4+\cdots+\log_N100\\\\=\log_N\{2.3.4...100\}\\\\=\log_N\{1.2.3.4...100\}\\\\=\log_N{100!}\\\\=\dfrac{1}{\log_{100!}N}\\\\=R.H.S.$

Hence proved.

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3 years ago

A projectile is shot into the air following the path,h(x) = - 5t2 + 20t + 3. At what time will it reach a maximum height? t = 1

kramer

Answer:

t = 2

Step-by-step explanation:

Notice that this expression for the projectile's path is that of a quadratic function with negative leading term. The graph of it therefore consists of a parabola with the branches pointing down (due to he negative leading coefficient). Therefore, the maximum of such parabola will reside at its vertex.

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