MAT 150 Module 9 – Logarithmic Functions Lesson 1 – Logarithmic functions and their properties Welcome to module 9 lesson 1 – Logarithmic functions and their properties.

Definition of a Logarithm The logarithmic function is the inverse of the exponential function. Instead of knowing the power, we are trying to find the power. Exponential function Y = ax Raise the base, a, to a power, x, to get the output, y. The logarithmic function, or Log function is the inverse of the exponential function with the same base. Knowing the inverse of an exponential function is used when we know the output of an exponential function (y) and we need work backwards to find the power (x). If f(x) = ax is an exponential function with base a, then its inverse is f-1(x) = logax, which is called “Log base a of x”. Logarithmic function Y = logax We know the output and we want to find the power, x.

Definition of a Logartithm Logax = ? is really asking a? = x Logax means, a raised to what power will equal x? so you are thinking, “what power would I have to raise “a” to in order to get x?”

Logarithmic form and exponential form ax = y Logarithmic Form logay = x ax = y is called exponential form and logay = x is called logarithmic form. The statements ax = y and logay = x mean exactly the same thing. This is very important to know! They are just two different ways of saying that the number a, raised to the x power, equals y.

Graphing Since the exponential function, y = ax, and the logarithmic function, y = logax, are inverses of each other, their graphs are symmetric across the line y = x. Here you can see the graphs of y = 2x and its inverse, y = log2x, which are symmetric across the line y = x. Note that you can graph logarithmic functions in desmos by clicking the “functions” menu, then “misc.”

Example 1 Rewrite in exponential form: logpy = B

Example 1 pB = y Rewrite in exponential form: logpy = B P is the base B is the exponent Y is the output To write the exponential function we identify that P is the base, B is the exponent, and y is the output. In exponential form, this would be written as pB = y. pB = y

Example 2 Rewrite in logarithmic form: w = r2

Example 2 logrw = 2 Rewrite in logarithmic form: w = r2 r is the base 2 is the exponent w is the output We identify that r is the base, w is the output, and 2 is the power. In logarithmic form, this would be written as logrw = 2. logrw = 2

Example 3 Evaluate without a calculator: Log2512 = Log 10,000 = For example 3, evaluate these logarithms without a calculator.

Example 3 - Solution Evaluate without a calculator: Log2512 = 9 -2 2 is the base and 512 is the output. Think: 2? = 512 29 = 512 Question 1, Log2512 = ?. 2 is the base and 512 is the output. Think, 2 raised to what power, is 512? 2 raised to the 9th power is 512, so Log2512 = 9. Question 2, Log5(1/25) = ?. 5 is the base and 1/25 is the output. Think, 5 raised to what power, is 1/25? 5 raised to the -2 power is 1/25, so Log5(1/25) = -2. Remember negative exponents create fractions, not negative numbers. 5 is the base and 1/25 is the output. Think: 5? = 1/25 5-2 = 1/25

Example 3 Evaluate without a calculator: 3. Log 10,000 =4 4.Log77 = 1. 10 is the base and 10,000 is the output. Think: 10? = 10,000 104 = 10,000 For Question 3, 10 is the base and 10000 is the output. If no base is written, the base is 10. Think: 10? = 10,000. 104 = 10,000 so Log 10,000 =4. For Question 4, 7 is the base and 7 is the output. Think: 7? = 7. 71 = 7 so Log77 = 1. Any time the base and output are the same, the power must be 1. That is, Logaa = 1 for any number a. 7 is the base and 7 is the output. Think: 7? = 7 71 = 7

Example 3 Evaluate without a calculator: 5. Log2 1= 0 6. Log7 -5 = Not possible 2 is the base and 1 is the output. Think: 2? = 1 20 = 1 Question 5, Log21 = 0. 2 is the base and 1 is the output. Think, 2 raised to what power, is 1? In order to raise a number to a power and get 1, the power must be 0. So Log21 = 0. This is true for any base: Loga1 = 0 since the only way to get an output of 1 is with a power of 0. Question 6, 7 is the base and -5 is the output. Think: 7? = -5. It is not possible to raise a positive number to a power and get a negative output. 7 raised to any power will always be a positive number. So Log7 -5 is not possible. The logarithmic function is only defined for positive numbers. 7 is the base and -5 is the output. Think: 7? = -5 7x cannot equal -5 for any power

When you see ln(x), think of it as Loge x. The Natural Logarithm The logarithmic function with base e is called the Natural Logarithm and is written ln(x). When you see ln(x), think of it as Loge x. The logarithmic function with base e is called the Natural Logarithm and is written ln(x).That’s lowercase l as in Lima. When you see ln(x), think of it as Loge x.

Evaluating Logs with a calculator Most scientific calculators have two buttons log Base 10 ln Base e Since the calculator only does base 10 (the log button) or base e (the ln button), we need something so we can evaluate logs with different bases on the calculator. We can evaluate logs with any base using the formula log base a of b = log b / log a. This is called the change of base formula. To calculate any other bases, use the change of base formula

Example 4 Evaluate with a calculator. Round to four decimal places. Log 75 = Ln 3.175 = Log215 = log579 = Evaluate these logarithms with a calculator. Round to four decimal places.

Example 4 - Solution  

Properties of Logarithms Logarithms have some interesting properties that allow us to simplify and rewrite them. Before calculators existed, these properties were used as a shortcut to do complicated multiplication and division problems. Logarithms have some interesting properties that allow us to simplify and rewrite them. Before calculators existed, these properties were used as a shortcut to do complicated multiplication and division problems.

Basic Properties of Logarithms . loga(a) = 1 since a1 = a for any a loga(1) = 0 since a0 = 1 for any a Basic Properties: loga(a) = 1 since a1 = a for any a. loga(1) = 0 since a0 = 1 for any a. loga(ax) = x since ax = ax for any a and any x and a^logax =x also since ax = ax for any a and any x loga(ax) = x since ax = ax for any a and any x since ax = ax for any a and any x

Basic Properties - Examples 1. Log66 = 1 since 61 = 6 2. Log201 = 0 since 200 = 1 3. Log552= 2 since 52 = 52. 4. = 26 since 326 = 326 Here are some examples using the basic properties.Log66 = 1 since 61 = 6 2. Log201 = 0 since 200 = 1 3. Log552= 2 since 52 = 52. 4. 3^Log326= 26 since 326 = 326

The Addition and Subtraction Properties Recall from what we know about exponents that : bM • bN = bM+N The Addition property of logarithms is: Recall from what we know about exponents that : bM • bN = bM+N. The Addition property of logarithms is: logb(MN) = logbM + logbN. logb(MN) = logbM + logbN

The Addition and Subtraction Properties Recall from what we know about exponents that : The subtraction property of logarithms is: Recall from what we know about exponents that : B^M/B^N = B^(M-N) The subtraction property of logarithms is: Logb(M/N) = logbM – logbN

Addition and Subtraction Properties – Example 1 Rewrite as the sum or difference of two logarithms. a. Log5(25x) = b. Log4 = Rewrite as the sum or difference of two logarithms with the same base. a. Log5(25x) = b. Log4 (P/Q)=

Addition and Subtraction Properties – Example 1 Solution Rewrite as the sum or difference of two logarithms. Using the addition property. Then we know Log5(25) = 2 since 52 = 25. Log5(25x) = Log5(25) + Log5(x) Rewrite as the sum or difference of two logarithms. Log5(25x) = Log5(25) + Log5(x) using the addition property. Then we know Log5(25) = 2 since 52 = 25. So Log5(25x) = 2 + Log5(x). Log5(25x) = 2 + Log5(x)

Addition and Subtraction Properties – Example 1 Solution Rewrite as the sum or difference of two logarithms. Log4 = Log4P - Log4Q using the subtraction property. Rewrite as the sum or difference of two logarithms. b. Log4(P/Q) = Log4P - Log4Q using the subtraction property.

The Power Property logaxn = n*logax This property tells us that logarithms turn exponents into coefficients: logaxn = n*logax This property tells us that logarithms turn exponents into multiplication: logaxn = n*logax

The Power Property – Example 2 Rewrite the following using the power property: Log4(x5) =   Ln(c7) = 8log6p = Rewrite the following using the power property:

The Power Property – Example 2 Solutions Rewrite the following using the power property: Log4(x5) =  5Log4(x) Ln(c7) = 7Ln(c) 8log6p = log6p8 Bring down the exponent of 5 and make it a coefficient. Bring down the exponent of 5 and make it a coefficient. Rewrite the following using the power property: Log4(x5) =  5Log4(x). Bring down the exponent of 5 and make it a coefficient. Ln(c7) = 7Ln(c). Bring down the exponent of 7 and make it a coefficient. 8log6p = log6p8 .Using the property in reverse, bring up the coefficient of 8 and make it an exponent. Using the property in reverse, bring up the coefficient of 8 and make it an exponent.

Using the properties together We can use any or all of the logarithms properties together to combine logarithms or rewrite them as a sum/difference of logarithms. We can use any or all of the logarithmic properties together to combine logarithms or rewrite them as a sum/difference of logarithms

Example 3 Use the properties of logarithms together to rewrite the expression. Log3(x2y3) = Log7 Use the properties of logarithms together to rewrite the expression.

Log3(x2y3) = Log3(x2) + Log3(y3) Example 3 Use the properties of logarithms together to rewrite the expression. Log3(x2y3) First, use the addition property since x2y3 is multiplied. Log3(x2y3) = Log3(x2) + Log3(y3) First use the addition property since x2y3 are multiplied. Turn the single logarithm into two that are added. Then use the power property to bring down the powers and make them coefficients. Then use the power property to bring down the powers and make them coefficients. Log3(x2) + Log3(y3) = 2Log3(x) + 3Log3(y)

Example 3 Use the properties of logarithms together to rewrite the expression. b. Log7 First, use the subtraction property since x5√m is divided by k. Log7[(x5√m)/k] = Log7(x5√m) - Log7(k) First, use the subtraction property since x5√m is divided by k. Next, use the addition property since x5√m is multiplied. Then use the power property to bring down the powers and make them coefficients. Note that the square root is the same as an exponent of ½. Next, use the additionproperty since x5√m is multiplied. = Log7(x5) + Log7(√m) - Log7(k) Then use the power property to bring down the powers and make them coefficients. = 5Log7(x) + ½Log7(m) - Log7(k)

Example 4 Use the properties of logarithms together to rewrite the expression. 2 Log2 w + 5 Log2 p = 5 ln (x) + 2 ln (y) - 3 ln (z)= Use the properties of logarithms together to rewrite the expression as one logarithm.

Example 4 - Solution Use the properties of logarithms together to rewrite the expression. 2 Log2 w + 5 Log2 p = First, use the power property to bring up the coefficients and make them powers. 2 Log2 w + 5 Log2 p = Log2 w2 + Log2 p5 First, use the power property to bring up the coefficients and make them powers. Next use the multiplication property to combine the two logs that are added into one where w and p are multiplied. Log2 w2 + Log2 p5 = Log2 (w2 p5) Next use the addition property to combine the two logs that are added into one where w and p are multiplied.

Example 4 Use the properties of logarithms together to rewrite the expression. 5 ln (x) + 2 ln (y) - 3 ln (z)= First, use the power property to bring up the coefficients and make them powers. = Ln(x5) + Ln(y2) – Ln(z3) First, use the power property to bring up the coefficients and make them powers. 5Ln(x) + 2Ln(y) – 3Ln(z) = Ln(x5) + Ln(y2) – Ln(z3) Next use the addition and subtraction properties to combine the three logs into one where x and y are multiplied and z is divided. Then we have Ln[(x5y2)/z3]. Next use the addition and subtraction properties to combine the three logs into one where x and y are multiplied and z is divided.