The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

The value of $\frac{\log_{3} 5 \times \log_{25} 27 \times \log_{49} 7}{\log_{81} 3}$ is

If $n = 1000!$,then $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + ... + \frac{1}{\log_{1000} n} = ......$

The roots of the equation $2^{x + 2} \cdot 27^{x/(x - 1)} = 9$ are given by

If $n = 1983!$,then the value of the expression $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \frac{1}{\log_4 n} + \dots + \frac{1}{\log_{1983} n}$ is equal to

If $\log _{7} 2 = \lambda$, then the value of $\log _{49} (28)$ is