Chapter 5 exponential and logarithmic functions 5 4 common and natural logarithmic functions
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Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions. Essential Question: What is the relationship between a logarithm and an exponent?. 5.4: Common and Natural Logarithmic Functions. You’ve ran across a multitude of inverses in mathematics so far...

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Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions

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Chapter 5 exponential and logarithmic functions 5 4 common and natural logarithmic functions

Chapter 5: Exponential and Logarithmic Functions5.4: Common and Natural Logarithmic Functions

Essential Question: What is the relationship between a logarithm and an exponent?


5 4 common and natural logarithmic functions

5.4: Common and Natural Logarithmic Functions

  • You’ve ran across a multitude of inverses in mathematics so far...

    • Additive Inverses: 3 & -3

    • Multiplicative Inverses: 2 & ½

    • Inverse of powers: x4 & or x¼

    • But what do you do when the exponent is unknown? For example, how would you solve 3x = 28, other than guess & check?

    • Welcome to logs…


5 4 common and natural logarithmic functions1

5.4: Common and Natural Logarithmic Functions

  • Logs

    • There are three types of commonly used logs

      • Common logarithms (base 10)

      • Natural logarithms (base e)

      • Binary logarithms (base 2)

    • We’re only going to concentrate on the first two types of logarithms, the 3rd is used primarily in computer science.

    • Want to take a guess as to why I used the words “base” above?


5 4 common and natural logarithmic functions2

5.4: Common and Natural Logarithmic Functions

  • Common logarithms

    • The functions f(x) = 10x and g(x) = log x are inverse functions

      • log v = u if and only if 10u = v

    • All logs can be thought of as a way to solve for an exponent

      • Log base answer = exponent

x

x

10

=

2

10

=

2

x

log

10


5 4 common and natural logarithmic functions3

5.4: Common and Natural Logarithmic Functions

  • Common logarithms

    • Scientific/graphing calculators have the logarithmic tables built in, on our TI-86s, the “log” button is below the graph key.

    • To find the log of 29, simply type “log 29”, and you will be returned the answer 1.4624.

      • That means, 101.4624 = 29

    • Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places


5 4 common and natural logarithmic functions4

5.4: Common and Natural Logarithmic Functions

  • Evaluating Common Logarithms

    • Without using a calculator, find the following

    • log 1000

    • log 1

    • log

    • log (-3)

If log 1000 = x, then 10x = 1000. Because 103 = 1000, log 1000 = 3

If log 1 = x, then 10x = 1 Because 100 = 1, log 1 = 0

If log (-3) = x, then 10x = (-3) Because there is no real number exponent of 10 to get -3 (or any negative number, for that matter), log(-3) is undefined


5 4 common and natural logarithmic functions5

5.4: Common and Natural Logarithmic Functions

  • Using Equivalent Statements (log)

    • Solve each by using equivalent statements (and calculator, if necessary)

    • log x = 2

    • 10x = 29

    • Remember

      • Log base answer = exponent

log x = 2 → 102 = x → 100 = x

10x = 29 → log 29 = x → 1.4624 = x


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5.4: Common and Natural Logarithmic Functions

  • Natural logarithms

    • (or Captain’s Log, star date 2.71828182846…)

      • Common logarithms are used when the base is 10.

      • Another regular base is used with exponents, that being the irrational constant e.

    • For natural logarithms, we use “ln” instead of “log”. The ln key is located beneath the log key on your calculator.


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5.4: Common and Natural Logarithmic Functions

  • Evaluating Natural Logarithms

    • Use a calculator to find each value.

      • ln 0.15

        • ln 0.15 = -1.8971, which means e-1.8971 = 0.15

      • ln 186

        • ln 186 = 5.2257, which means e5.2257 = 186

      • ln (-5)

        • Undefined, as it’s not possible for a positive number (e) to somehow yield a negative number.


5 4 common and natural logarithmic functions8

5.4: Common and Natural Logarithmic Functions

  • Using Equivalent Statements (ln)

    • Solve each by using equivalent statements (and calculator, if necessary)

    • ln x = 4

    • ex = 5

    • Remember

      • Log base answer = exponent

ln x = 4 → e4 = x → 54.5982 = x

ex = 5 → ln 5 = x → 1.6094 = x


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5.4: Common and Natural Logarithmic Functions

  • Assignment

    • Page 361, 2 – 36 (even problems)

      • Even problems are done exactly like the odd problems, which are in the back of the book)


Chapter 5 exponential and logarithmic functions 5 4 common and natural logarithmic functions day 2

Chapter 5: Exponential and Logarithmic Functions5.4: Common and Natural Logarithmic FunctionsDay 2

Essential Question: What is the relationship between a logarithm and an exponent?


5 4 common and natural logarithmic functions10

5.4: Common and Natural Logarithmic Functions

  • Graphs of Logarithmic Functions


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5.4: Common and Natural Logarithmic Functions

  • Transforming Logarithmic Functions

    • Same as before…

      • Changes next to the x affect the graph horizontally and opposite as would be expected

      • Changes away from the x affect the graph vertically and as expected

      • Example

        • Describe the transformation from the graph of g(x) = log x to the graph of h(x) = 2 log (x – 3). Give the domain and range.

          • Vertical stretch by a factor of 2

          • Horizontal shift to the right 3 units

          • Domain: The domain of a log function is all positive real numbers (x > 0). Shifting three units right means the new domain is x > 3.

          • Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.


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5.4: Common and Natural Logarithmic Functions

  • Transforming Logarithmic Functions

    • Example #2

      • Describe the transformation from the graph of g(x) = ln x to the graph of h(x) = ln (2 – x) - 3. Give the domain and range.

        • x is supposed to come first, so h(x) should be rewritten as h(x) = ln [-(x – 2)] - 3

        • Horizontal reflection

        • Horizontal shift to the right 2 units

        • Vertical shift down 3 units

        • Domain: The domain of a log function is all positive real numbers (x > 0). The horizontal reflection flips the sign, and shifting two units right means the new domain is x < 2.

        • Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.


5 4 common and natural logarithmic functions13

5.4: Common and Natural Logarithmic Functions

  • Assignment

    • Page 361, 37 – 48 (all problems)

      • Problems 37 – 40 only ask to find the domain, but you may need to figure out the translation first.

      • Even problems are done exactly like the odd problems, which are in the back of the book)


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