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Fundamental Laws of Logarithms

log2 , log(2^...

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\log2 , \log (2^n-1 ) , \log( 2^n+3) are in ap then n=

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a_{1}, a_{2}, a_{3}, . . .

are in GP then the values of the determinant

∣ ∣ ∣ ∣ \begin{matrix} l o g a_{1} & log a_{n + 1} & log a_{2 n + 1} l o g a_{2} & log a_{n + 2} & log a_{2 n + 2} l o g a_{3} & log a_{n + 3} & log a_{2 n + 3} \end{matrix} ∣ ∣ ∣ ∣

log 2

log (2^{x} - 1)

log (2^{x} + 3)

are in AP , then the value of

x

Q. If in an AP,

t_{1} = {log}_{10} a

t_{n + 1} = {log}_{10} b

t_{2 n + 1} = {log}_{10} c

a, b, c

f (x) = lim n \to \infty \frac{log (2 + x) - x^{2 n} sin x}{1 + x^{2 n}}

Q. Assertion :When

| x | < 1, lim n \to \infty \frac{log (x + 2) - x^{2 n} c o s x}{x^{2 n} + 1} = log (x + 2)

- 1 < x < 1

n \to \infty, x^{2 n} \to 0

.

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