Answer:
0.197
Step-by-step explanation:
Given equation:
[tex]-9.3e^{10b}=-67[/tex]
To solve the equation, begin by dividing both sides by -9.3:
[tex]\dfrac{-9.3e^{10b}}{-9.3}=\dfrac{-67}{-9.3}\\\\\\e^{10b}=\dfrac{670}{93}[/tex]
Take the natural logs of both sides:
[tex]\ln e^{10b}=\ln \left(\dfrac{670}{93}\right)[/tex]
[tex]\textsf{Apply the natural log power law:} \quad \ln x^n=n\ln x[/tex]
[tex]10b\ln e=\ln \left(\dfrac{670}{93}\right)[/tex]
As ln(e) = 1, then:
[tex]10b=\ln \left(\dfrac{670}{93}\right)[/tex]
Divide both sides by 10 to isolate b:
[tex]\dfrac{10b}{10}=\dfrac{\ln \left(\dfrac{670}{93}\right)}{10}\\\\\\b=\dfrac{1}{10}\ln \left(\dfrac{670}{93}\right)[/tex]
Evaluate:
[tex]b=0.197467821923...\\\\\\b=0.197\; \sf (nearest\;thousandth)[/tex]
Therefore, the value of b rounded to the nearest thousandth (3 decimal places) is:
[tex]\LARGE\boxed{\boxed{0.197}}[/tex]
