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