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