AA3-4 Investigation -- intro to logs cloned 1/19/2021

So far in this unit we have looked at how to write and solve an exponential equation for a future value.  

For example,  AA3-3 Hypothetical #1:  
          The last day of school was March 13.  On that day, Multnomah County had 2 carriers.  The infection rate 
          was 1.85.  This was about 32 weeks ago.  If that infection rate continued, calculate, how many people 
          would become infected this week.

          We wrote the equation f(x) = 2(1.85)x  then found f(32) ≈ 708,800,000

          Then we used the graph to work backwards and determine how many weeks it would be until 825,000 
          people would be infected in one week.  The answer was about 21 weeks.

     

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Try this one. If you borrow $500 on a credit card charging 24% annual (yearly) interest and do not make payments, how much would you owe after 4 years? Hint: Credit card debt is exponential growth.

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

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You should have learned it was $1182.11. The calculator gave you this number.

What if we wanted to find out how long it would take to owe $5,000?

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

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Using the graph you should have seen that it would take about 10.7 years.

The issue: How can we solve this without graphing? 500(1.24)x = 5000

Step 1: Isolate the grouping symbol. (1.24)x = _______

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To undo the multiply by 500, you should have divided by 500. Division is the inverse of multiplying

How do we solve (1.24)x=10
What is the inverse operation?

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To isolate an exponent, we need a new inverse operation. It is called a logarithm (we call them "logs").

This is the definition of a logarithm:

The base is written as a subscript.

How would you describe the steps in rewriting an exponent as a log to someone who couldn't see the definition?

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In our equation (1.24)x=10, what is the base?

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Use the definition of a logarithm to rewrite (1.24)x=10, as a log.

You will need to use the equation editor.

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Rewrite 100=5x as a log.

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Rewrite 2x=14 as a log.

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Once you have it written as a log, the next thing you need to know is how to enter it in a calculator. This can be the tricky part.

The method to solve a log depends on the calculator you are using. In Desmos below you need to use the equation editor (the keyboard button).

Then choose functions on the right side.
Then choose Misc.
Then choose loga

solve log 1.24 (10) = x

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

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What answer did you get? (Round to the nearest 100th.)

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On an iPhone it can be more confusing. You have to get to the logy button (it is in the sideways, 2nd menu).
You type the base, logy, then what you are taking the log of.

In our example you would type, in order: 1.24 logy 10 =

The LOG button on most calculators does not give you the option of typing in a base. Those calculators have a default base of 10. For this we have to use the base change rule.

This changes any base to two logs with base 10. This rule is good to know because it will work on any calculator and can be easier to enter.
You can enter into desmos without even using the functions button. Try it below:

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

1

Use a calculator or desmos below to solve:

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

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x = _______

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Use a calculator or desmos below to solve:

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

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x=

1

Putting it all together:

On March 13, Multnomah County had 2 COVID 19 carriers. The infection rate was 1.85. If that infection rate continued,how many weeks would it be until 100,000 people would be infected in one week?

We have released a new and improved Graphing question type! Students will no longer be able to answer this question.

1

Try this one. If you borrow $500 on a credit card charging 24% annual (yearly) interest and do not make payments, how much would you owe after 4 years? Hint: Credit card debt is exponential growth.

You should have learned it was $1182.11. The calculator gave you this number. What if we wanted to find out how long it would take to owe $5,000?

Using the graph you should have seen that it would take about 10.7 years. The issue: How can we solve this without graphing? 500(1.24)x = 5000 Step 1: Isolate the grouping symbol. (1.24)x = _______

To undo the multiply by 500, you should have divided by 500. Division is the inverse of multiplyingHow do we solve (1.24)x=10What is the inverse operation?

To isolate an exponent, we need a new inverse operation. It is called a logarithm (we call them "logs"). This is the definition of a logarithm:The base is written as a subscript. How would you describe the steps in rewriting an exponent as a log to someone who couldn't see the definition?

In our equation (1.24)x=10, what is the base?

Use the definition of a logarithm to rewrite (1.24)x=10, as a log. You will need to use the equation editor.

Rewrite 100=5x as a log.

Rewrite 2x=14 as a log.

What answer did you get? (Round to the nearest 100th.)

Use a calculator or desmos below to solve:

x = _______

Use a calculator or desmos below to solve:

x=

Putting it all together: On March 13, Multnomah County had 2 COVID 19 carriers. The infection rate was 1.85. If that infection rate continued,how many weeks would it be until 100,000 people would be infected in one week?

How many weeks?

You should have learned it was $1182.11. The calculator gave you this number.

What if we wanted to find out how long it would take to owe $5,000?

Using the graph you should have seen that it would take about 10.7 years.

The issue: How can we solve this without graphing? 500(1.24)x = 5000

Step 1: Isolate the grouping symbol. (1.24)x = _______

To undo the multiply by 500, you should have divided by 500. Division is the inverse of multiplying

How do we solve (1.24)x=10
What is the inverse operation?

To isolate an exponent, we need a new inverse operation. It is called a logarithm (we call them "logs").

This is the definition of a logarithm:

The base is written as a subscript.

How would you describe the steps in rewriting an exponent as a log to someone who couldn't see the definition?

Use the definition of a logarithm to rewrite (1.24)x=10, as a log.

You will need to use the equation editor.

Putting it all together:

On March 13, Multnomah County had 2 COVID 19 carriers. The infection rate was 1.85. If that infection rate continued,how many weeks would it be until 100,000 people would be infected in one week?