LOGARITHMS Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics

You should be able to…  Use logarithms to solve exponential equation with different bases  Apply the laws of logarithms to simplify expressions  Find a logarithm to any base by using the change of base formula  Solve equations involving logarithms

Investigate 1. a) The number of bacteria in a culture starts at 1, and doubles every hour. Let y represent the number of bacteria, and x represent the time, in hours. Make a table of values relating x and y. b) What exponential function relates y to x? 2. To determine when there will be 15 bacteria in the culture, what equation would have to be solved?

Investigate continued… 3. a) Graph y = 2 x and its inverse on the same set of axes. Recall that, to find the inverse function, interchange the x- and y-values. b) Describe the graphical relationship between a function and its inverse, and their relationship to the line y = x. c) Explain how the graph of the inverse can be used to approximate the solution to the equation determined in step 2.

Investigate continued… 4. Find an approximate solution for each equation by carefully graphing an appropriate function and its inverse. a) 3 x = 12 b) 6 x = 17 c) –2 x = –9 5. Make a general statement explaining how to solve exponential equations graphically.

How can you solve for an unknown exponent? Suppose you invest $100 in an account that pays 5% interest, compounded annually. The amount A, in dollars, in the account after any given time, t, in years, is given by.  Predict how long it will take, to the nearest year, for the amount in this account to double in value. Give reason for your estimate.

 Design a method that will allow you to find an accurate answer to your estimation.  Carry out your method. How long will it take for the investment to double in value?  Compare this result with your prediction. How close was your predication?

 Suppose the initial amount invested is $250. How does this affect your answer from the last step? Explain, using mathematical reasoning. Reflect – How can you express the original equation in logarithmic form?

What is the power law for logarithms? Evaluate each logarithm. Organize your results in a table. (i) log 2 (ii) log 4 (iii) log 8 (iv) log 16 (v) log 32 Look for a pattern in your results. How are these vales related to log 2? Make a prediction for (i) log 64 (ii) log 1024 Verify your predictions

Write a rule for the general result of log 2 n  Do you think the general result will work for other powers?  Repeat the analysis for powers of 3: (i) log 3 (ii) log 9 (iii) log 27  Write a rule for evaluating log 3 n.  Verify your rule using a few cases.

Reflect  Write a rule for evaluating  Test your rule using several different cases.

Task – Not Fatal Viral infections, while often quite severe and requiring admission to the hospital, are rarely fatal. Most people recover and go home. In this task, you will simulate recovery from a viral infection for 100 people who contract the infection. Materials needed: Heads represents a person who has recovered, and tails represents a person who is still ill.

Task – Not Fatal   Toss 100 coins.  Count all the coins that turn up heads. These are the people who have recovered after one week in the hospital, and get to go home.  Record in a table of values the number of coins remaining (people still ill) versus the number of tosses.  Repeat the process with the remaining coins until you have no coins left.

Task – Not Fatal  Use technology to construct a scatterplot of the data, with the number of tosses (days) as the independent variable and the number of people still ill as the dependent variable.  Determine an equation for the curve of best fit for your data. Explain how you determined the best model.  Predict how ling it would take for 1600 people to get well and go home.  Justify your prediction algebraically.  Why does a logarithmic model work well for this situation?  For what other situations might this model be appropriate? Justify your answer.

Question Given that log 5 x = y, express each of the following in terms of y. (a) log 5 x 2

Question continued… Given that log 5 x = y, express each of the following in terms of y. (b)

Question continued… Given that log 5 x = y, express each of the following in terms of y. (c) log 25 x

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Practice

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Practice

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Practice

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You should know…  How to think of logarithms as exponents—that is, log a x means “the exponent to which a must be raised to give x”. For example, log 3 9 means “the exponent to which 3 must be raised to give 9”, which is 2. Therefore, log 3 9 = 2.  The laws of logarithms:

You should know…  If it is not possible to make the bases the same, you can use the properties of logarithms to solve an exponential equation. For example:

Be prepared…  One of the keys to success in solving logarithmic equations is your ability to move easily between the logarithmic form and the exponential form. Remember that if a = log b x, then x = b n.  The change of base formula allows you to express any logarithm in terms of another base. This is useful when graphing logarithmic functions or evaluating logarithms that do not have base 10 or base e.