# Thread: Exponential and Logarithmic Equations

1. ## Exponential and Logarithmic Equations

Good Morning,

My problem is: Use a calculator with a y^x key or ^ key to solve the following.

The 1986 explosion at Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The functon f(x)=1000(0.5)^ x over 30 describes the amount, f(x), in kilograms of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyls atmosphere, the area is considered unsafe for human habitation. Find f(80) and determine if Chernobyl will be safe for human habitation by 2066.

2. Originally Posted by kbryant05
Good Morning,

My problem is: Use a calculator with a y^x key or ^ key to solve the following.

The 1986 explosion at Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The functon f(x)=1000(0.5)^ x over 30 describes the amount, f(x), in kilograms of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyls atmosphere, the area is considered unsafe for human habitation. Find f(80) and determine if Chernobyl will be safe for human habitation by 2066.
Hello,

I hope that you are allowed to use the ln-button too.

Plug in the value(s) you know into the equation:

$f(x)=1000 \cdot \left( \frac{1}{2} \right)^{\frac{x}{30}$

That means you have to solve the equation for x:

$80=1000 \cdot \left( \frac{1}{2} \right)^{\frac{x}{30}$

The next steps should be:

$0.08=\left( \frac{1}{2} \right)^{\frac{x}{30}$

The exponent x/30 is a logarithm to the base 0.5. Use the base-change-formula (I am not sure, if this is the right expression. But maybe you recognize the name when you have a look at the solution)

$\frac{\ln(0.08)}{\ln(0.5)}=\frac{x}{30}$

So:

$30\cdot\frac{\ln(0.08)}{\ln(0.5)}=x$. That means x is round about 110 years. In 2066 Chernobyl will be one of the most dangerous places on earth.

Greetings

EB

3. ## Reply on problem

Thanks Earboth. I don't think that answer is very comforting for anyone living in that area. It's actually really frightening.
Yes we are allowed to use the in button too!!
Have a great day!

4. Originally Posted by earboth
That means x is round about 110 years. In 2066 Chernobyl will be one of the most dangerous places on earth.

Greetings

EB
I don't agree. You said that $f(x)=f(80)=80$, which would solve the equation for "how many years does it take to reach 80kg" but the question says it is $f(x)=f(80)$ so $x=80$

So you would substitute 80 for x instead of the function of x.

$f(x)=1000(0.5)^{\frac{x}{30}}$

$f(80)=1000(0.5)^{\frac{80}{30}}$

$f(80)=1000(0.5)^{\frac{8}{3}}$

$f(80)=1000(0.157490131236859)$

$f(80)=157.490131236859$

So, yes it is still lethal.