Find the exact value of [loge 1]+[loge 2]+[loge 3]+…+[loge 66]

Name: Problem 23, Chapter 1: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2020-05-11T07:13:10-08:00
Duration: 43 s
Channel: Jeyasree R T
Description: Find the exact value of [loge 1]+[loge 2]+[loge 3]+…+[loge 66]

step 1 We need to find the exact value of the given logarithm without using a calculator. The given log

Find the exact value of [loge 1]+[loge 2]+[loge...

Key Concepts

Floor Function (Greatest Integer Function)

The floor function maps a real number to the largest integer less than or equal to that number. In problems that involve summing floor values of functions, it is essential to understand how the function's output is 'binned' into integer intervals, thereby allowing the summation to be broken into sums over distinct intervals where the floor remains constant.

Natural Logarithm

The natural logarithm, denoted as log_e, is the logarithm to the base e. It serves as the inverse function to the exponential function. In this context, its properties, such as its continuous, monotonically increasing behavior, and its specific values at e^k, are used to determine the intervals over which the floor function remains constant.

Interval Partitioning Using Logarithmic Boundaries

By analyzing the natural logarithm function, one can partition the domain of integers into intervals defined by the inequality e^k ? n < e^(k+1) for integers k. Within each interval, the floor of the logarithm remains constant. This method allows for efficient summing of the floor values by multiplying the constant floor value by the number of integers in the interval.

Counting Integers in an Interval

Once the intervals where the floor function is constant are established, it is necessary to count the integers within each interval. This counting process is crucial for converting the problem of summing a function at each integer into a simpler problem involving multiplication of the constant floor value by the count of integers in that interval.

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