Exponential and Logarithmic Equations and Modeling Problems

[jacksonpeeble]

Hello everyone! I have two precalculus problems that are from an assessment today that I had some great trouble with. I wrote the objectives below, followed by the problems themselves. Any help is greatly appreciated!

Homework Statement

1. Solve exponential and logarithmic equations when possible. For those that cannot be solved analytically, use graphic methods to find approximate solutions.

2. Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay.

Homework Equations

1. Given log_bx=2, log_by=3, and log_bz=-2, find log_b((x²*z)/y³)

2. The half-life of radium-226 is 1600 years. Suppose you have a 22mg sample. After how long will only 18mg of the sample remain?

The Attempt at a Solution

These were the two that I was completely stumped on. I do recall that the formula for half-lives is m(t)=m₀e^-n where r=ln2/h.

 

[grief]

1.

Remember the properties of logarithms.

for all x,y,n,
log(x^n) = n*log(x)
log(x*y)=log(x)+log(y)
log(x/y)=log(x)-log(y)

these should help you convert your expression into a simplified expression involving only log(x) log(y) and log(z).

2. Halflife means the number of years that it takes the substance to decompose to exactly half of its original mass. But of course the substance is not suddenly decompose in half. It's a continuous process.

Think of it this way. If your substance has half of its original mass, 1600 years have gone by. If it has a fourth of its original mass, another 1600 years have gone by. If you have (1/2)^n, 1600*n years have gone by. But n need not be an integer. In your problem, 18/22 of the original substance remains. For what real number n is 18/22 equal to (1/2)^n? If you know what the definition of logarithm is, you should know how to solve this.

 

[jacksonpeeble]

So, if I'm correct, I need to sort of reverse the process...

log_b((x²*z)/y³)

I'm really still pretty lost on this one. I've been working at this type of problem for hours. I don't want just the answer; I already had the quiz. However, the test is coming up and I need to know how to solve these.

The key thing throwing me off is how I'm supposed to plug in the values for the final equation. A step-by-step would really be great.

For the halflife problems, I think I have the solution and understand, but I'd appreciate if somebody could double-check my answer...

1600=(-ln(2))/k)
k=(-ln2)/1600
k=~-0.000433216988

N=N₀e^k*t
18=22*e^{-0.000433216988*t}
t=~463.210587512

 

[grief]

log_b(x²*z/y³)=log_b(x²)+ log_bz-log_b(y³)= 2log_bx +log_bz -3log_by = 2*2+(-2)-3*3=-7.

You're right about the half-life problem.

 

[jacksonpeeble]

Thank you for your help, grief!

 

[Mark44]

So, if I'm correct, I need to sort of reverse the process...

log_b((x²*z)/y³)

I'm really still pretty lost on this one. I've been working at this type of problem for hours. I don't want just the answer; I already had the quiz. However, the test is coming up and I need to know how to solve these.

The key thing throwing me off is how I'm supposed to plug in the values for the final equation. A step-by-step would really be great.

For the halflife problems, I think I have the solution and understand, but I'd appreciate if somebody could double-check my answer...

1600=(-ln(2))/k)
k=(-ln2)/1600
k=~-0.000433216988

N=N₀e^k*t
18=22*e^{-0.000433216988*t}
t=~463.210587512

Don't forget to include the units in your answer. Also, the data you were given only has two significant digits, so the extra precision in your approximation is unwarranted.

 

[jacksonpeeble]

Don't forget to include the units in your answer. Also, the data you were given only has two significant digits, so the extra precision in your approximation is unwarranted.

Thanks for the tip, Mark. Oddly, this is for Honors Trigonometry and Precalculus, not Honors Chemistry, so we're supposed to round to three decimal places (AP standards). I do often forget to label, though, so thanks again!

Just to clarify, had it been Honors Chemistry, should I have rounded to two significant figures total, or two decimal places in?

 

[Mark44]

Thanks for the tip, Mark. Oddly, this is for Honors Trigonometry and Precalculus, not Honors Chemistry, so we're supposed to round to three decimal places (AP standards). I do often forget to label, though, so thanks again!

Just to clarify, had it been Honors Chemistry, should I have rounded to two significant figures total, or two decimal places in?

Two significant figures. The data you were given (22 mg, 18 mg, 1600 years) all had only two significant digits. If you're uncertain about significant figures, here's a Wikipedia article on this subject: http://en.wikipedia.org/wiki/Significant_figures
Mark