Exponential Functions & Logarithms  Name:   

Notes  Date:   

 

 

Jorge is working with a client who has received an inheritance of                       $50,000. The client would like to put the money into mutual funds                       and wants to diversify the investments. Jorge makes 30 year                   portfolio estimates using the expected annual rate of return for                   “Best Case” and “Worst Case” scenarios. Jorge must use his model                     to determine the projections for each case. 

 

 

 Write the definition of the term and include an image or example that represents it. 

Term  Definition  Example 

Exponential Functions 

  

Logarithm    

Natural Logarithm 

  

Exponential Growth 

  

Exponential Decay 

  

Half-Life    

Doubling Time 

  

  

 

What is an Exponential Function?  The Conceptualizer! 

Exponential functions are functions that have a             positive number as base and the power is the                 variable. It is usually in the form of ,                ay = x  where .a > 0   One special characteristic of an exponential           function is the change of the variable has a                 much greater impact on the value of the               function as compared to other functions where             the variable is the base rather than the               exponent.  This is where the common term “exponential growth” come from. This is a term describing a growth that starts slow but increases very rapidly.  A linear function is one that is changing at a                   constant rate as x changes. An exponential             function is one that changes at a rate that's                 always proportional to the value of the             function.  Exponential functions have important uses in           science, economics and population.    Recall! Negative exponents means switching the base           with the exponent to the denominator.  In this case, when x decreases, the value of the exponential function will be approaching zero. 

 

 

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What is an Exponential Function?  The Conceptualizer! 

Generally speaking, the graph of an           exponential function is either strictly         increasing or strictly decreasing. looks like a             flat line on one end and steep slope on the                   other end.   Often, due to the steepness of an exponential               function, the scale of the y-axis is compressed               compared to the scale of the x-axis.  

 

 

 

Solving Simple Exponential Equations  Details 

To solve simple exponential equations, we can             make sure both sides are of the same base.                 After we have the same base, the two powers                 on the two sides must be equal and then we                   can solve the equation accordingly.    You must make sure you apply the exponents               rules and property correctly while changing           base. 

 

 

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Solving Exponential Equations 

without Logarithms Notes 

0805 · 6n−8 = 1    

 

 

Solving Exponential Equations 

without Logarithms Notes 

8x+6 = 23x · 43x   

 

 

Converting Between  

Exponential and Logarithms The Conceptualizer! 

A simple exponential equation involves a           base, an exponent and a result.   

Logarithms are the inverses of         exponentials and they help us to find the               exponents if given the result and the             base. They answer the question, “What           exponent gives us this result?”   

All exponential equations can be         converted to logarithmic equations and         vice versa. 

⇔ log ybx = y b = x  

 

 

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Converting Between Exponential and Logarithms  Answers 

Rewrite as a logarithmic equation.40173 = 2    

Rewrite as a logarithmic equation.0242x = 1    

Rewrite as a logarithmic equation.95mn = 1    

Rewrite as an exponential equation. 4096log4 = 6    

Rewrite as an exponential equation. 625log5 = x    

Rewrite as an exponential equation. 200logb = a    

 

 

Solving Exponential Equations with Logarithms  The Conceptualizer! 

Some exponential equations cannot be solved by             changing the bases. Therefore, we can take the               log (natural log or base-10 log) on both sides.nl   One important logarithm property states: 

(x ) xlogb y = y logb    Like with addition, subtraction, multiplication         or division, taking the log on both sides will not                   affect the balance of the equation. It will help                 us to “bring down” the exponent and solve the                 equation accordingly.   

 

 

 

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Solving with Natural Logarithms  The Conceptualizer! 

If the base is e, we will make use of natural log                       instead of base-10 log.   There is not anything else that changes with the                 way the problem is solved.  

 

 

 

Solving Exponential Equations with Logarithms  Notes 

79t = 6     

 

 

Solving with Natural Logarithms  Notes 

08e2n−1 = 3     

 

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Growth and Decay  The Conceptualizer! 

We investigated how the value of x             can affect the value of the           exponential function .y a = x    When , the function will always  a = 1          be 1 as whatever power we raised 1 to                 we will end up with 1.  When , as x increases, thea > 1          function is multiplied by a whole           number (the base a) so its value will               be magnified. Therefore, the function         will be increasing (more and more           rapidly) as x increases. In this case,             we say the function models growth.  When , the function is  a < 1        multiplied by a fraction less than one             (a) its value will be smaller than the               original number. Therefore, the       function will be decreasing (more and           more rapidly) as x increases. In this             case, we say the function models           decay.  What if the exponent in the function             is negative?   Consider , where a > 1y a = −x   

a y = −x = 1ax = ( a1)x  

 Since the base is a fraction less than               one, the function models      y a = −x  decay.  Similarly, if the base of an           exponential function is less than 1           with a negative exponent ,        y = ( a1)−x  the function models growth because         

. y = ( a1)−x = 1a−x = ax  

 

 

 

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 Write examples of exponential equations that model growth or decay. 

Growth  Decay 

   

 

 

Application of Growth and Decay  The Conceptualizer! 

The exponential growth or decay can be modeled by                 e P t = P 0rt  

 

where is the initial valueP 0    is the value at time P t t   is a growth/decay rater   ( for growth and for decay) r > 0 r < 0   is the timet   

The common applications of exponential growth and             decay are population (of a town or bacteria) or                 decay of radioactive substance.  

Usually, you are provided with all the information               except the one variable you are solving for. 

 

 

 

Application of Growth and Decay  Notes 

Due to a plague, there was an exponential decay in population of a city from the year of 1800 to                                       1850. The population in the year of 1800 was 4850 and it was down to 1500 in the year of 1850.                                         Find the decay rate. 

   

 

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Half-Life and Doubling Time  The Conceptualizer! 

Half-life is the time required for a quantity               to decrease in half in exponential decay.  Doubling time is the time required for a               quantity to double in exponential growth.  When we are given the half-life/doubling           time, we can find the rate of growth and                 decay even without the information of the             initial amount.   You can also find the half-life/doubling           time with just the rate of decay/growth.  In the case of half-life: where h          .5 P PH = 0 0    is the half-life.  In the case of double time: where            P PD = 2 0  D is the double time. 

 

 

 

Half-Life and Doubling Time  Notes 

If a community of rabbits is growing 14% monthly, what is the doubling time of its population? 

   

 

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What are the steps to the solving an exponential growth or decay problem? List them in order. 

Step 1   

Step 2   

Step 3   

Step 4   

 

 

Graphing Exponential Functions  Details 

Similar to graphing other functions, we will             need to find a few points that satisfy the                 exponential functions before graphing it.         However, the values of the exponential           functions can end up being very large or               very small and you might not find many               “plottable” points.   Therefore, the basic knowledge of what the             exponential functions do will help us a lot               when graphing.  Remember, the graph of an exponential           function looks like a flat line on one end                 and is a steep slope on the other end.  

 

 

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Graphing Exponential Functions  Notes 

Graph 3y = −x   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 Jenna buys a new car for $24,000. It depreciates at the rate of 15% per year. How long does it take                                         the car to lose half of its value?        Account A starts with $900 and earns 2.5% compound interest. Account B starts with $1000 and                               earns 1.5% compound interest. Which is worth more after 10 years?        Joey estimates that there are 1500 bacteria in his sample. The population doubles every hour. How                               long will it take to reach 20,000 bacteria?     

 

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