Quiz1) Convert log24 = x into exponential form
2) Convert 3y = 9 into logarithmic form
3) Graph y = log4x
2x = 4 log39 = y
y = log4x
Properties of LogarithmsWith logs there are ways to expand and condense them using properties
Product Property:loga(c*d) = logac + logad
Examples:log4(2x)
log8(x2y4)
= log42 + log4x
= log8x2 + log8y4
Division (Quotient) Property:loga(c/d) = logac – logad
Examples:log4(2/x)
log8(x2/y4)
= log42 – log4x
= log8x2 – log8y4
When two numbers are multiplied together within a log you can split them apart using separate logs connected with addition
When two numbers are divided within a log you can split them apart using separate logs connected with subtraction
Properties of Logarithms (continued)
Power Property:loga(cx) = x*logac
Examples:log4(x2)
log8(2x)
= 2log4x
= xlog82
Examples using more than one propertylog3(c2/d4)
log4(5x7)
log8((4x2)/y4)
= log45 +log4x7
= (log84 + log8x2) – log8y4
When a number is raised to a power within a log you multiply the exponent to the front and multiply it by the log (bring the exponent out front)
= 2log3c – 4log3d
= (log84 + 2log8x) – 4log8y
= log3c2 – log3d4
= log45 +7log4x
log9(63*210)
= 3log96 + 10log92
= log963 + log9210
Try These
Log1/2(4-3*5(2/3))
= -3log1/24 – (2/3)log1/25
= log1/24-3 – log1/25(2/3)
log3((1/2)3/(-2)-4)
= 3log3(1/2) – -4log3(-2)
= log3(1/2)3 – log3(-2)-4
= 3log3(1/2) + 4log3(-2)
Quiz1) Find: log5125
2) What two numbers would log424 be between?
5? = 125 51 = 5 52 = 25 53 = 125
41 = 4 42 = 16 43 = 64
So log5125 = 3
So log424 is between 2 and 33) Use a calculator to find log424
log424 = (log(24))/(log(4)) = 2.929
Condensing logarithms (undoing the properties)
= log5(6/y)
log95 + 7log9x
log56 – log5y
log212 – (7log2z + 2log2y)
= log95 + log9x7
= log9(5x7)
= log212 – (log2z7 + log2y2)= log212 – (log2(z7y2))
= log2(12/(z7y2))
Solve for x
Since the base is the same we can set the pieces that we are taking the log of equal to each other.
log525 = 2log5x
log4x = log42
25 = x2
We use the properties to condense the log – then solve for x
log525 = log5x2
5 = x
x = 2
Try These
log36 = log33 + log3x
6 = 3x log36 = log3(3x)
2 = x3 3
(1/3)log4x = log44
x(1/3) = 4 log4x(1/3) = log44
x = 64
(( ))3 3