Chapter 11.4, Problem 33WE

To determine

To show:

  ${\displaystyle \sum _{k=1}^{4}k\log 5=\log \left(5^{10}\right)}$

Explanation of Solution

Given information:

  ${\displaystyle \sum _{k=1}^{4}k\log 5=\log \left(5^{10}\right)}$

Proof:

Take a left hand side of the given equation:

  $\begin{array}{l}{\displaystyle \sum _{k=1}^{4}k\log 5}=\left(1\right)\log 5+\left(2\right)\log 5+\left(3\right)\log 5+\left(4\right)\log 5\\ {\displaystyle \sum _{k=1}^{4}k\log 5}=\left(10\right)\log 5\\ {\displaystyle \sum _{k=1}^{4}k\log 5}=\log 5^{10}\left[\because a\log x=\log x^a\right]\\ {\displaystyle \sum _{k=1}^{4}k\log 5}=\log \left(5^{10}\right)\end{array}$

Therefore, left hand side of the given equation equal to the right hand side.

Hence proved.