Chapter 4.4, Problem 44E

To determine

To calculate: The expression to use the laws of logarithm, $\log \left(\frac{{10}^x}{x\left(x^2+1\right)\left(x^4+2\right)}\right)$

Answer to Problem 44E

The expended form is, $x-8\log x-\log 2$ ,

Explanation of Solution

Given information:

Expend the expression, , $\log \left(\frac{{10}^x}{x\left(x^2+1\right)\left(x^4+2\right)}\right)$

Formula used:

For change the base rule,

  $\begin{array}{l}{\log }_{a}b=\frac{\log b}{\log a},\\ {\log }_a{b}^x=x{\log }_{a}b\end{array}$

Calculation :

Consider the expression, $\log \left(\frac{{10}^x}{x\left(x^2+1\right)\left(x^4+2\right)}\right)$

Rewrite the expression,

  $\begin{array}{c}\log \left(\frac{{10}^x}{x\left(x^2+1\right)\left(x^4+2\right)}\right)=\log \left(\frac{{10}^x}{\left(x^3+x\right)\left(x^4+2\right)}\right)\\ =\log {10}^x-\log \left(x^3+x\right)\left(x^4+2\right)\\ =x-3\log x-\log x-4\log x-\log 2\end{array}$

Apply the log base rule: , ${\log }_a\left(x\right)=\frac{{\log }_b\left(x\right)}{{\log }_b\left(a\right)},{\log }_a{b}^x=x{\log }_{a}b$

  $x-3\log x-\log x-4\log x-\log 2=x-8\log x-\log 2$

The above expression can’t be simplified further.

Thus the simplified form of the expression is, $x-8\log x-\log 2$ .