Properties of logs
The basic property of logarithims
• Loga(bc)=logab+Logac
Example
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Loga(b4)
=loga(bbbb)
=loga(b)+Loga(bbb)
=loga(b)+Loga(b) +Loga(b) +Loga(b)
=4Loga(b)
The basic properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
Example
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x=log832 what is x?
Rewrite as an exponential equation
8x=32
Take log2 of both sides
Log2(8x)=Log232
xLog2(8)=Log232
x=Log2(32)/Log2(8)
x=5/3
Change of base
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x=logay what is x?
Rewrite as an exponential equation
ax=y
Take logc of both sides
Logc(ax)=Logcy
xLogc(a)=Logcy
x=Logc(y)/Logc(a)
logay=Logc(y)/Logc(a)
Change of base
• Using this rule on your calculator
• logay=Logc(y)/Logc(a)
If you’re looking for the logay use…
Log(y)÷Log(a)
Or
ln(y)÷ln(a)
The basic properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
Warning: Remember order of operations
WRONG
log(2ax)
=x*log(2a)
=x*[log(2)+log(a)]
=x log 2 + x log a
CORRECT
log(2ax)
=log(2(ax))
=log(2)+log(ax)
=log(2)+x*log(a)
What about division?
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Loga(b/c)
=Loga(b(1/c))
=Loga(bc-1)
=Loga(b) + Loga(c-1)
=Loga(b) + -1*Loga(c)
=Loga(b) - Loga(c)
The advanced properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
 Loga(b/c)=logab-Logac
What about roots?
log a ( n b )
1/n
log a (b )
1
log a (b)
n
The advanced properties of logarithims
• Loga(bc)=logab+Logac
• Loga(bn)=n*logab
• Logab=logc(b)/logc(a)
 Side effect: you only ever need one log button on
your calculator.
 Logab=log(b)/log(a)
 Logab=ln(b)/ln(a)
 Loga(b/c)=logab-Logac
 Loga(n√b̅)=[logab]/n
REVIEW QUESTION
Simplify: log4(1/256)
a)
b)
c)
d)
e)
-2.40824
-5.5452
-.25
4
-4
REVIEW QUESTION
Simplify: log4(1/256)
There are lots of ways to do this. Here’s how I did it.
log4(1/256)
=log4(1)-log4(256)
=0-log4(256)
=0-log4(28)
=0-8*log4(2)
=0-8*log4(√4)
=0-8*log4(4½)
=0-8*(½)
=-4
E
Simplifying log expressions
Expand
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Log(3x4/√y)
=Log(3)+Log(x4)-Log(√y)
=Log(3)+Log(x4)-Log(y½)
=Log(3)+4*Log(x)-½Log(y)
• Step 1: Expand * into + and ÷ into –
• Step 2: convert nth roots into 1/n powers
• Step 3: Expand ^ into *
Condense
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Log(7)-2Log(x)+¾Log(q)
Log(7)-Log(x2)+Log(q¾)
Log(7/x2)+Log(q¾)
Log(7q¾/x2)
• Step 1: Condense * into ^
• Step 2: Condense – into ÷
• Step 3: Condense + into *
Condense using the properties of logarithms:
3log(x) -2log(y)
a)
b)
c)
d)
e)
3 2
log(x y )
3
2
log(x ) log(y )
3
2
log(x )/ log(y )
3
2
log(x - y )
None of the above
Condense using the properties of logarithms:
3log(x) -2log(y)
3log(x) -2log(y)  Condense * into ^
log(x3) -log(y2)  Condense – into ÷
log(x3/y2)
 E. None of the above.
Solving Exponential Equations
And log equations, too, I guess
Example
Solve:
Add 7 to both sides
3x+2 - 7 = 12
3
x+2
= 19
Take the log of both sides log(3x+2 ) = log(19)
Pull out the exponent
(x+2)log(3) = log(19)
Divide both sides by log(3) x+2 = log(19) / log(3)
Subtract 2 from both sides x=log(19)/log(3)-2
Use calculator if you want x » 0.680144
General Strategy
• An exponential equation has a variable in the
exponent
• Get the exponent part by itself
• Take the log() of both sides
– Or if you want, take the ln() of both sides
• Use properties of logs to pull the power out.
• Solve for your variable
Example with ln()
Solve:
Divide both sides by 4
Take the ln() of both sides
Pull the exponent out
ln(e)=1
Solve for x
4e 2 x-7 = 12
e 2 x-7 = 3
ln(e 2 x-7 ) = ln(3)
(2x - 7)ln(e) = ln(3)
2x - 7 = ln(3)
x = [ln(3) + 7] / 2
Solving log equations
Example
Solve:
log3 (2x + 7) + log3 (4) = 2
Combine logs
log3 (4(2x + 7)) = 2
log3 (8x + 28) = 2
Distribute
Exponentiate both sides 3log3 (8x+28) = 32
Cancel inverse functions 8x + 28 = 32
Solve for x
8x + 28 = 9
x = (9 - 28) / 8
x = -2.5
General strategy
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Combine logs to get one log by itself
Exponentiate both sides with the matching base
Exponential and log functions will cancel
Solve for x
Solve:
a)
b)
c)
d)
e)
x =5/e
x = ln(e)
x = ln(5)
x=5
None of the above
Solve:
ln(ex)=ln(5)
x*ln(e)=ln(5)
x*1=ln(5)
x=ln(5)
C