2 What is a logarithm?A logarithm is the power to which a number must be raised in order to get some other numberFor example, the base ten logarithm of 100 is 2, because ten raised to the power of two is 100:100 = 102 because log = 2
4 Remember: If y = bx then logby = x Ex: Write the following equations in logarithmic formRemember: If y = bx then logby = xIf 25 = 52thenLog525=2If 729 = 36thenLog3729=6If 1 = 100thenLog101=0Ifthen
5 Let’s try some: Converting between the two forms. Expo Form Log Form
6 Let’s pause for a second . . . If y = bx then logby = xx in the exponential expression bx is the logarithm in the equation logby=xThe base b in bx is the same as the base b in the logarithmNOTE: b does not =1 and must be greater than 0The logarithm of a negative number or zero is undefines.
7 Common LogsA common log is a logarithm that uses base 10. You can write the common logarithm log10y as log yScientists use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance. The pH of a substance equals
8 Evaluating Logarithms Ex: Evaluate log816Log816=x Write an equation in log form16 = 8x Convert to exponential form24 = (23)x Rewrite using the same base. In this case, base of 224 = 23x Power of exponents4 = 3x Set the exponents equal to each otherx=4/ Solve for xTherefore, Log816=4/3
9 Evaluating Logarithms Ex: EvaluateWrite an equation in log formConvert to exponential formRewrite using the same base. In this case, base of 2. Use negative expos!-5 = 6x Set the exponents equal to each otherx=-5/ Solve for xTherefore,
12 Graphs of Logarithmic Functions A logarithmic function is the inverse of an exponential functionIn other words, y= 10x and y=log10x are inverses of each other.Where is the line of reflection?y = 10xY = log10x
13 Let’s try a more complicated one Find the inverse of y=log5(x-1)+2y=log5(x -1) Start with the original functionx=log5(y -1) Switch the x and yx-2=log5(y -1) Subtract 2 from both sidesy-1=5(x-2) Rewrite in y=abx formy=5(x-2) Add 1 to both sidesThe inverse of y=log5(x-1)+2 is y=5(x-2)+1
14 Let’s try some Find the inverse of each function: Y=log0.5x y=log5x2 y=log(x-2)Hint: what is the base?
15 Solutions Find the inverse of each function: Y=log0.5x y=log5x2 y=log(x-2)
16 Now, make a table of values: 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=log2x?y=2xNow, make a table of values:
17 Graph of y=log2x and y=2x Make a table of values and graph y=2x X y=2x -30.125-20.25-10.512438What is the domain and range of this function?D: all real numbers. R: y>0
18 Graph of y=log2x and y=2xNow, reverse the coordinates to graph y=log2xPrediction: what will it look like?xy=2x0.125-30.25-20.5-112483What is the domain and range of this function?D: x>0. R: all real numbers
19 Graph of y=log2x and y=2x Graphing both functions, we get this. Where is the line of reflection?y=x
20 Let’s try one:Graph y=log3(x+3). Determine the domain and range.
24 Determining the domain and range Domain: x>0Range: all real numbersy=log2xDomain: x>-3Range: all real numbersy=log3(x+3)How can you predict the domain and range without graphing?Write a sentence summarizing how to determine domain and range.
25 Predict: what is the domain and range of y=log3(x-2) and y=log3(x)+1 y=log3(x-2) y=log3(x)+1Domain: x>0Range: all real numbersNote: the +1 moves the parent graph up but does not affect the left or right translation, so the domain remains x>0Domain: x>2Range: all real numbers