Chapter 11.6, Problem 16CFU

To determine

To find: the solution of the Bach equation or inequality by graphing and round solutions to the nearest hundredth.

Answer to Problem 16CFU

The nearest hundredth value is $24.46$ .

Explanation of Solution

Given:

  $5^{x+3}>{10}^{x-6}$ .

Concept used:

According the logarithmic formula:

  $\begin{array}{l}{\log }_{10}x=2.303\ln x.\\ \log e=1.\\ \log e^a=a\log e=a.\\ \log x^a=a\log x.\end{array}$

The value of the logarithm can be calculated from calculator or logarithmic table.

Calculation:

According to the given:

  $5^{x+3}>{10}^{x-6}$ .

Taking logarithm on both sides:

  $\begin{array}{l}\log 5^{x+3}>\log {10}^{x-6}.\\ \Rightarrow \left(x+3\right)\log 5>\left(x-6\right)\log 10.\\ \Rightarrow \left(x+3\right)0.669=\left(x-6\right)1.\\ \Rightarrow 0.699x+2.097=x-6.\end{array}$

By continue the above equation:

  $\begin{array}{l}6+2.097>x-0.669x.\\ \Rightarrow 8.097>0.331x.\\ x>\mathrm{24.46.}\end{array}$

Hence, the nearest hundredth value is $24.46$ .