Word problem-population with logs

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In summary, the conversation is about finding the population formulas for two countries, UDAB and FILO, and using logarithms to solve for when UDAB's population will reach 75,000 people. The formula for UDAB is P_{U}(t)=5(1.055)^t and the formula for FILO is P_{F}(t)=3t+60=3(t+20). To solve for when UDAB's population reaches 75,000, we use logs to get an approximate answer of 7.573 years.
  • #1
jibbs
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Hi, I've been staring at this problem for days and still cannot figure out how to begin:

In 1980 the population of UDAB was 50,000 people and since was increasing continuously by 5.5% per year for the next 30 years. On the other hand the population of FILO was 60,000 and increasing by 3,000 people per year over the same time period. For each country, write a formula expressing the population as a function of time, where t is the number of years since 1980. In how many years will the population of UDAB reach approximately 75,000 (use logs to solve).

I have in formula for UDAB as a=pert, A=50,000e^.055(30). I cannot figure out a formula for FILO. And while I can figure out how to solve the second part without logs, I am lost on how to do with using logs.

Any help would be greatly appreciated!
 
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  • #2
I think I would express the population of UDAB as (in tens of thousands):

\(\displaystyle P_{U}(t)=5(1.055)^t\)

And since the population of FILO increases by the same amount each year, I would write (the population in thousands):

\(\displaystyle P_{F}(t)=3t+60=3(t+20)\)

Then to answer the second part of the problem, we could write the equation:

\(\displaystyle P_{U}(t)=7.5\)

\(\displaystyle 5(1.055)^t=7.5\)

Now, solve for $t$...what do you find?
 
  • #3
We have:

\(\displaystyle 5(1.055)^t=7.5\)

Divide through by 5:

\(\displaystyle (1.055)^t=\frac{3}{2}\)

Convert from exponential to logarithmic form:

\(\displaystyle t=\log_{1.055}\left(\frac{3}{2}\right)\)

Use change of base formula to get an expression we can use to obtain a decimal approximation using a calculator:

\(\displaystyle t=\frac{\ln\left(\dfrac{3}{2}\right)}{\ln(1.055)}\approx7.573\)
 

FAQ: Word problem-population with logs

What is a word problem-population with logs?

A word problem-population with logs is a type of mathematical problem that involves using logarithms to solve for the population of a certain group or area. It typically involves using real-world data and applying mathematical equations to determine the population size.

How do you solve a word problem-population with logs?

To solve a word problem-population with logs, you first need to identify the given information and what you are trying to solve for. Then, you can use the formula for population growth or decay, which involves taking the logarithm of both sides of the equation. Finally, you can plug in the given values and use the properties of logarithms to solve for the unknown variable.

What are some common mistakes when solving word problems-population with logs?

Some common mistakes when solving word problems-population with logs include not using the correct formula, forgetting to take the logarithm of both sides of the equation, and making errors when applying the properties of logarithms. It is important to double-check your work and make sure all steps are correct before submitting your final answer.

How does solving word problems-population with logs relate to real-world scenarios?

Solving word problems-population with logs is a valuable skill because it allows us to understand and analyze real-world data. Many real-world scenarios involve populations that are constantly changing, and using logarithms can help us make predictions and projections about these populations. It also helps us understand the rate at which a population is growing or declining.

Are there any tips for solving word problems-population with logs?

Some tips for solving word problems-population with logs include carefully reading the problem and identifying the given information, using the correct formula, and checking your work for errors. It is also helpful to practice and become familiar with the properties of logarithms and how to apply them in different scenarios. Additionally, drawing a diagram or making a table can help organize the information and make the problem easier to solve.

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