4A: #1e-l,2-5,6g-l (Base 10 logs) 4B: #1-3last col,4,5,6 last col (Base a logs)

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1 1. Understand the meaning of and use logs of base 10 and base a 4A: #1e-l,2-5,6g-l (Base 10 logs) 4B: #1-3last col,4,5,6 last col (Base a logs) We have been looking at exponential functions of the form f(x) = a b (x - c) + d. A specific example might be the amount of money in a bank as a function of time: A(t) = 1500(1.0125) 4t We have explored certain questions all of which involved a known time t. But what if we were to ask how long does it take for the amount to reach 3000? We can start by writing the equation: 3000 = 1500(1.0125) 4t Dividing both sides by 1500 gives: 2 = (1.0125) 4t But this creates a troubling question: What do we have to raise to in order to get 2? If we knew, we could divide that by 4 to get our answer. Let's look at a simpler version of this question. What is the solution to 10 x = 100? This is easy since we can write 100 as 10 2 and equate exponents: x = 2! Suppose we want to solve 10 x = 50. We know that x has to be bigger than 1 but less than 10. What is it exactly? We give it a name: Definition of Base 10 Logarithm If 10 x = a then x = log 10 a...or... log10a is the exponent to which you raise 10 to get a. Notice that the log function undoes the exponentiation. It is the inverse of the exponential function. Graphically, it is the reflection around the line y = x (like all inverses) Geogebra Demo Notice that: Base 10 logs are particularly useful since we work in a decimal system. Consider the following logs: Base 10 is so common that if the base is absent, it is assumed to be base 10! So what would log(1235) be? Your calculator will give you an exact answer, but you should know that it's between 3 (10 3 = 1000) and 4 (10 4 = 10,000) and thus, it is closer to 3. How about log(55,472)? Let's work with these some more:

2 b We've seen logs in base 10. The same idea holds for any base. In other words: Geogebra Demo Definition of Logarithm If b x = y then x = log b y...or... logb y is the exponent to which you raise b to get y. Notice that: and... Equivalent equations can be written in logarithmic or exponential form. For example 3 x = 12 means x = log312 Try some: Write equivalent equations for the following: Evaluate some logs for situations you know: Another question: What is the log 2(-32)? The domain of any log function must be restricted to positive arguments! 4A: #1e-l,2-5,6g-l (Base 10 logs) 4B: #1-3last col,4,5,6 last col (Base a logs) The most common bases, other than 10 and e are 2 and 3. As such, it's important to know by memory some of the lower powers of 2 and 3. You know the powers of 2 up to Now learn the powers of 3 through 3 5.

3 Invitation: UNM-PNM Friday, 11/4. Register at UNM-PNM web site. Powers of 10 4A: #1e-l,2-5,6g-l (Base 10 logs) 4B: #1-3last col,4,5,6 last col (Base a logs) Discuss a few from #6. Take questions Understand and use laws of logarithms 4C.1: #1col3,2cfi,3def,4-7last row (Laws of logs) 4C.2: #1-2lastcol,3,4lastcol (Log equations) QB: #4,5,6a,7ab,13 (QB Log Properties) (due Tue, 11/1) Every log equation has an equivalent Let's look at some properties of logarithms: exponential version - and vice versa! What would the log of a product be? That is, can we rewrite log(ab)? Let log(a) = x and log(b) = y Then, 10 x = a and 10 y = b Multiplying these two means 10 x 10 y = ab or 10 x + y = ab Using the definition, log(ab) = x + y or log(ab) = log(a) + log(b) Hmmm... think you could do somethings similar to find a rule for? Let log(a) = x and log(b) = y Then, 10 x = a and 10 y = b Dividing these two means 10 x 10 y = or 10 x - y = Using the definition, x - y = or = log(a)- log(b) Hmmm...another common situation is to evaluate log a b Any ideas? Let x = log a Then, by definition, 10 x = a Raise both sides to the b to get (10 x ) b = a b or 10 xb = a b log(a b ) = bx Since x = log a, we can rewrite this as log(a b ) = b log a More generally, these results hold for any base c > 0 as long as c 1: And a reminder: from the definition, we have the following properties Properties of Logarithms Let's see some applications: Write as a single logarithm:

4 .2 1. Solve equations involving logarithms by equating arguments Consider the following equations involving logarithms. Can you solve them? A reminder about the relationship between powers and logarithms: Since log bx is the power to which we raise b to get x, we have: Similarly, logbb x is the power to which we raise b to get b x so: Properties of Logarithms These can help simplify our lives: For example, is just log 3(3-4/5 ) So it evaluates to -4/5. Try a few: /4 Calculating logs that are not known. The log key on your calculator calculates log base 10 or the common logarithm The ln key on your calculator calculates log base e or the natural logarithm We will discuss how to find logs of other bases soon. First, let's look at how to write some equations in logarithmic form. As we do these, think about how we might work backward from the answer to the original equation. Consider y = 2 x. Take the log base 10 of both sides to get log(y) = log 2 x or log(y) = xlog2 Or how about? Can you write it as a log base 10 equation? log(r) = log(b) log(l) Try. log(l) = log(a) + log(b) - log(c) Note that we're using log 10 above. We can use any base (some are more useful than others) You can take the logarithm of both sides of an equation and use the rules of logarithms to rewrite an equation in logarithmic form. Now try to try to remove the logarithms from some equations by using the properties of logarithms: Finally, let's use these ideas to solve some logarithmic equations: Alternatively, What was the strategy? We used log rules to get a single log of the same base on both sides What was the strategy? We simplified into a single log on one side then used definition of log. Alternatively, Solving equations with logs 1) Manipulate both sides until there is one log term of the same base on each side. 2) Set the arguments of the two log functions equal. 3) Solve for the variable Note: Working with logarithmic equations can create extraneous solutions because the domain of the log function is restricted! Check your answers! 4C.1: #1col3,2cfi,3def,4-7last row (Laws of logs) 4C.2: #1-2lastcol,3,4lastcol (Log equations) QB: #4,5,6a,7ab,13 (QB Log Properties) (due Tue, 11/1)

5 4C.1: #1col3,2cfi,3def,4-7last row (Laws of logs) 4C.2: #1-2lastcol,3,4lastcol (Log equations) Present 2f,3f,4f,5f,6f Present 1h,2dfh,3,4df Quick Quiz 1. Write as a single logarithm or integer: [2 marks each] a) b) c) 2. [2 marks each] a) b) 3. Write the equations without logarithms: [2 marks each] a) b) 4. Solve the equations: [3 marks each] a) b).1&2 1. Strengthen log skills working with natural logs 4D.1: #1-4all,5-6lastrow (Natural logarithms) 4D.2: #1-4lastcol,5efgh (Laws of natural logs (not new, more practice)) Natural logarithms are logarithms with a base of the natural number e. They follow the same rules as any other logarithm. They are very common and loge is written as ln. Scientific calculators have a ln key on them Some specific identities for natural logs include: Section D.2 is more practice with the laws of logarithms, this time using natural logs. Do you recall how to work with equations that have constants in them? Write the constant as a log of the base to the constant. 4D.1: #1-4all,5-6lastrow (Natural logarithms) 4D.2: #1-4lastcol,5efgh (Laws of natural logs (not new, more practice))...and QB: #4,5,6a,7ab,13 (QB Log Properties) (due Tue,11/1)

6 We will have the Unit Test on Exponents and Logarithms on Tuesday, 11/8 4D.1: #1-4all,5-6lastrow (Natural logarithms) 4D.2: #1-4lastcol,5efgh (Laws of natural logs (not new, more practice)) QB: #4,5,6a,7ab,13 (QB Log Properties) Questions only Present: #2i,3f,4f,5h Present: all QB Quick Quiz 1. Write as a single logarithm or integer [2] [2] 2. Show that [2] [2] 3. Write without logarithms 4. Solve for x [2] [2] 1. Solve equations involving logarithms by equating arguments 4E: #1-2lastcol,3,5-6deh,7bc (Natural log equations) 4F: #1-3abc,4 (Change of base formula) QB: #3,7c (QB Change of Base) We have looked at setting exponents equal to solve exponential equations. But the bases on both sides have to be the same. When that is not possible, you can take the log (use any convenient base) of both sides of an equation. Some examples: Solving equations with exponents when the bases are different! 1) Simplify both sides as much as possible first. 2) Take the log of both sides, using an appropriate base depending on the situation. 3) Manipulate the equation with log rules until you can isolate the variable. 4) Simplify the result, using a calculator with proper rounding or as an exact answer.

7 1. Understand and apply the change of base formula Consider the equation y = 2 x. From the definition of logarithms, x = log2y. But you can also solve this, as we've done, by taking the log of both sides using any base (10 is convenient): log y = log 2 x Using the log rule for powers, we get: log y = xlog 2 Dividing both sides by log 2: Hmmm... There was nothing special about using base 10. So in general: Change of Base Formula for Logarithms Use whatever base is convenient for c. A couple constraints: a, b, c > 0 and b& c 1 Here's another way to see it: Try a couple: Estimate without a calculator log 5 14,500 log 54, log /0.8 6 Here's one in a context Your calculator will do this: [MATH][A:LogBASE] or [ALPHA][WINDOW][5:LogBASE] Caution: Overuse of calculators can lead to arthritis of the fingers, blisters, drowsiness and the inability to text. Unit Test on Exponents and Logarithms will be Tuesday, 11/8 4E: #1-2lastcol,3,5-6deh,7bc (Natural log equations) 4F: #1-3abc,4 (Change of base formula) QB: #3,7c (QB Change of Base)

8 Unit Test on Exponents and Logarithms will be Tuesday, 11/8 4E: #1-2lastcol,3,5-6deh,7bc (Natural log equations) Present #5d,6d, 7c 4F: #1-3abc,4 (Change of base formula) Present #1c,2c,3c,4 1. Graph and interpret graphs of transformations of log functions 4G: #1-2ad,3,4,6 (Graphs of logarithms) What does a graph of a logarithm look like. Recall that logarithms are inverses of exponentials. Can you graph the function f(x) = log(x)? Let's look at these in more detail. GGB Exponentials & Logs Properties of Graphs of Logarithms The graph of y = log b(x) has the following properties for any base b: The curve is a reflection of a corresponding exponential across the line y = x (they are inverses) The domain is x > 0 (we can only find logs of positive numbers) The graph has a vertical asymptote at x = 0 (the y-axis) The domain of log b(g(x)) is given by the values of x that make g(x) > 0 Consider the family of graphs that are generated by transforming y = log b(x) by scaling and shifting the curve. The generalized form of the family of log functions is: y = a log b(x - c) + d In a way that is analogous to exponential functions, we have: Summary of Graphs of Logarithmic Curves For f(x) = a log b(x - c) + d a stretches the curve vertically > a > 0 stretches the curve > 0 < a < 1 compresses the curve > a < 0 reflects over the x-axis (flips vertically) b increases or decreases the rate of change (steepness) of the curve > b > 1 curve is increasing > 0 < b < 1 curve is decreasing > b < 0 is for a future math class c shifts the curve horizontally > c > 0 shifts the curve to the right > c < 0 shifts the curve to the left d shifts the curve vertically > d > 0 shifts the curve up > d < 0 shifts the curve down Try one. Sketch a graph of Hint: A good sketch will have at least 2 points labelled, indicate axis intercepts, and show relevant asymptotes (as equations, of course). Unit Test on Exponents and Logarithms will be Tuesday, 11/8 4G: #1-2ad,3,4,6 (Graphs of logarithms)

9 4G: #1-2ad,3,4,6 (Graphs of logarithms) Discuss questions and #3,6 1. Solve problems involving growth and decay 4H: #2-14 even (Solve growth & decay using logs) QB: #8,9,10,11,16 (QB Growth & Decay) Very common application of exponents and logarithms involve real world growth and decay. Let's look at some examples: Use your calculator here... 4H: #2-14 even (Solve growth & decay using logs) QB: #8,9,10,11,16 (QB Growth & Decay) This completes our work with logarithms. We will have an exam on exponents and logarithms next time.