Chapter 3.4, Problem 10E

Solution

To write: The given expression as a sum or difference of logarithms or multiples of logarithms.

The given logarithm expression $\log 1000x^4$ is written as $3\log 2+3\log 5+4\log x$ .

Given information:

The given logarithm expression is $\log 1000x^4$ .

Concept Used:

Let $b.R,$ and $S$ be positive real number with $b\ne 1$ , and $c$ any real number.

Product Rule: ${\log }_{b}RS={\log }_{b}R+{\log }_{b}S$

Quotient Rule: ${\log }_b\frac{R}{S}={\log }_{b}R-{\log }_{b}S$

Power Rule: ${\log }_b{R}^c=c{\log }_{b}R$

Calculation:

Use the product rule of the logarithm, ${\log }_{b}RS={\log }_{b}R+{\log }_{b}S$ .

  $\log 1000x^4=\log 1000+\log x^4$

Write $1000$ as ${\left(2\cdot 5\right)}^3$ .

  $\log 1000+\log x^4=\log {\left(2\cdot 5\right)}^3+\log x^4$

Use the power rule of the logarithm, ${\log }_b{R}^c=c{\log }_{b}R$ .

  $\log {\left(2\cdot 5\right)}^3+\log x^4=3\log \left(2\cdot 5\right)+4\log x$

Use the product rule of the logarithm, ${\log }_{b}RS={\log }_{b}R+{\log }_{b}S$ .

  $\begin{array}{c}3\log \left(2\cdot 5\right)+4\log x=3\left(\log 2+\log 5\right)+4\log x\\ =3\log 2+3\log 5+4\log x\end{array}$

Hence, the expanded form of the logarithm expression is $3\log 2+3\log 5+4\log x$ .