Properties of logs

The basic property of logarithims

• Loga(bc)=logab+Logac

Example

• 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

• 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

• 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)Orln(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

WRONGlog(2ax)=x*log(2a)=x*[log(2)+log(a)]=x log 2 + x log a

CORRECTlog(2ax)=log(2(ax))=log(2)+log(ax)=log(2)+x*log(a)

What about division?

• 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?

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 QUESTIONSimplify: log4(1/256)

a) -2.40824b) -5.5452c) -.25d) 4e) -4

REVIEW QUESTIONSimplify: 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

• 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

• 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) log(x3y2)

b) log(x3) log(y2)

c) log(x3)/ log(y2)

d) log(x3 - y2)

e) 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

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()

Solving log equations

Example

General strategy

• 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: 5xe

a) x =5/eb) x = ln(e)c) x = ln(5)d) x = 5e) None of the above

Solve: 5xe

ln(ex)=ln(5)x*ln(e)=ln(5)x*1=ln(5)x=ln(5) C