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8.3 – Logarithmic Functions and Inverses

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What is a logarithm? A logarithm is the power to which a number must be raised in order to get some other number For example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100: 100 = 10 2 because log = 2

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UP, DOWN, UP

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Ex: Write the following equations in logarithmic form Remember: If y = b x then log b y = x If 25 = 5 2 then Log 5 25=2 If 729 = 3 6 then Log 3 729=6 If 1 = 10 0 then Log 10 1=0 If then

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Let’s try some: Converting between the two forms. Expo Form Log Form

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Let’s pause for a second... If y = b x then log b y = x x in the exponential expression b x is the logarithm in the equation logb y =x The base b in bx is the same as the base b in the logarithm NOTE: b does not =1 and must be greater than 0 The logarithm of a negative number or zero is undefines.

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Common Logs A common log is a logarithm that uses base 10. You can write the common logarithm log 10 y as log y Scientists use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance. The pH of a substance equals

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Evaluating Logarithms Ex: Evaluate log 8 16 Log 8 16=x Write an equation in log form 16 = 8 x Convert to exponential form 2 4 = (2 3 ) x Rewrite using the same base. In this case, base of = 2 3x Power of exponents 4 = 3x Set the exponents equal to each other x=4/3 Solve for x Therefore, Log 8 16=4/3

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Evaluating Logarithms Ex: Evaluate Write an equation in log form Convert to exponential form Rewrite using the same base. In this case, base of 2. Use negative expos! -5 = 6x Set the exponents equal to each other x=-5/6 Solve for x Therefore,

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Let’s try some Evaluate the following:

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Let’s try some Evaluate the following:

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Graphs of Logarithmic Functions A logarithmic function is the inverse of an exponential function In other words, y= 10 x and y=log 10 x are inverses of each other. Where is the line of reflection? y = 10 x Y = log 10 x

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Let’s try a more complicated one Find the inverse of y=log 5 (x-1)+2 y=log 5 (x -1)+2 Start with the original function x=log 5 (y -1)+2 Switch the x and y x-2=log 5 (y -1) Subtract 2 from both sides y-1=5 (x-2) Rewrite in y=ab x form y=5 (x-2) +1 Add 1 to both sides The inverse of y=log 5 (x-1)+2 is y=5 (x-2) +1

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Let’s try some Find the inverse of each function: Y=log 0.5 xy=log 5 x 2 y=log(x-2) Hint: what is the base?

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Solutions Find the inverse of each function: Y=log 0.5 xy=log 5 x 2 y=log(x-2)

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How does this help us? We don’t have a way to graph a log, BUT we can graph an exponential function. If we can find the inverse of the log function and graph it, the graph of the log is simply the reflection of the exponential function. Let’s start with an easy one: what is the inverse of y=log 2 x? y=2 x Now, make a table of values:

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Graph of y=log 2 x and y=2 x Make a table of values and graph y=2 x Xy=2 x What is the domain and range of this function? D: all real numbers. R: y>0

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Graph of y=log 2 x and y=2 x Now, reverse the coordinates to graph y=log 2 x xy=2 x Prediction: what will it look like? What is the domain and range of this function? D: x>0. R: all real numbers

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Graph of y=log 2 x and y=2 x Graphing both functions, we get this. Where is the line of reflection? y=x

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Let’s try one: Graph y=log 3 (x+3). Determine the domain and range.

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Let’s try one: Graph y=log 3 (x+3)

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Let’s try one: Graph y=log 3 (x+3)

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Let’s try one: Graph y=log 3 (x+3)

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Determining the domain and range Domain: x>0 Range: all real numbers y=log 2 x y=log 3 (x+3) Domain: x>-3 Range: all real numbers How can you predict the domain and range without graphing? Write a sentence summarizing how to determine domain and range.

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y=log 3 (x-2) y=log 3 (x)+1 Predict: what is the domain and range of y=log 3 (x-2) and y=log 3 (x)+1 Domain: x>2 Range: all real numbers Domain: x>0 Range: all real numbers Note: the +1 moves the parent graph up but does not affect the left or right translation, so the domain remains x>0

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