Chapter EP, Problem 8.6.17EP

To determine

To find: The equation or inequality and round to four decimal places.

Answer to Problem 8.6.17EP

The equation or inequality and round to four decimal places is $\left\{t|t<11.9929\right\}$ .

Explanation of Solution

Given information:

The given equation is ${11}^{t+1}\le {22}^{t+3}$ .

Calculation:

Consider the equation.

Let b and x be positive numbers $b\ne 1$ . The logarithm of x with base b is denoted by ${\log }_{b}x$ and is defined as the exponent y that makes the equation $b^y=x$ true.

Consider the following properties of logarithms,

Calculate the expression.

${\log }_b\frac{m}{n}={\log }_{b}m-{\log }_{b}n$

Product properly:

${\log }_{b}mn={\log }_{b}m+{\log }_{b}n$

Power property:

${\log }_b{m}^p=p{\log }_{b}m$

Consider the following inequality:

${11}^{t+1}\le {22}^{t+3}$

Apply property of inequality for logarithmic functions:

$\log {11}^{t+1}\le \log {22}^{t+3}$

Apply power property of logarithm:

$(t+1)\log 11\le (t+3)\log 22$

Apply distributive property $\left(\text{a}+\text{b}\right)\text{c}=\text{ac}+\text{bc}$

$t\log 11+\log 11\le t\log 22+3\log 22$

Subtract $\text{log11}$ from both sides:

$t\log 11-\log 22\le 3\log 22-\log 11$

Again apply distributive property:

$t(\log 11-\log 22)\le 3\log 22-\log 11$

Again apply power property of logarithm:

$t(\log 11-\log 22)\le \log 10648-\log 11$

Divide both the sides by $(\log 11-\log 22)$ :

$t\le \frac{\log 10648-\log 11}{(\log 11-\log 22)}$

Use a Ti-83 calculator to solve above value:

Therefore, the equation or inequality and round to four decimal places is $t\le 11.9929$ .