Chapter EP, Problem 8.6.18EP

To determine

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

Answer to Problem 8.6.18EP

The equation or inequality and round to four decimal places is $a=2.946865$ .

Explanation of Solution

Given information:

The given equation is $2^{3a-1}=3^{a+2}$ .

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,

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

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:

$2^{3a-1}=3^{a+2}$

Take log both sides

$\log 2^{3a-1}=\log 3^{a+2}$

Apply power property of logarithm:

$(3a-1)\log 2=(a+2)\log 3$

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

$3a\log 2-\log 2=a\log 3+2\log 3$

Subtract $a\log 3$ from both sides:

$3a\log 2-a\log 3-\log 2=2\log 3$

Add log2 in both sides:

$3a\log 2-a\log 3=2\log 3+\log 2$

Again apply distributive property:

$a(3\log 2-\log 3)=2\log 3+\log 2$

Again apply power property of logarithm:

$a(\log 8-\log 3)=\log 9+\log 2$

Divide both the sides by $(\log 8-\log 3)$ :

$a=\frac{\log 9+\log 2}{(\log 8-\log 3)}$

Use a Ti-83 calculator to solve above value:

Therefore, the equation or equality round to four decimal is $a=2.946865$