name: solutions& * * * …eckertsc1.weebly.com/uploads/5/8/4/6/58466249/u1_test_2...log18 log6...
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PRE$CALCULUS* * * * ** * *TEST*2*Review*and*study*guide**KEY*EXPONENTIAL*AND*LOGARITHMIC*FUNCTIONS* * * * !!Write&the&log&equation&as&an&exponential&equation:&&!"#!256 = !& & &
Then&solve&for&x.& & & & & & &
4! = 256.&&&
You&can&see&that&256&is&the&same&as&4!.&&So&4! = 4!, !"!! = 4.&&OR,&from&the&original&logarithmic&equation&
you&can&use&change&of&base&formula&to&show&! = ! !"# !"#!"# ! = 4.&&
Use&the&properties&of&logs&to&expand:&&!"# !!!!
!& &
This&is&the&same&as&log !!!!
!!.&&Then&
!! [log !
! − log!!].!!!ℎ!"ℎ!!ℎ!"!!"#$%"&! !! [4 log ! − 5 log!].&&
Simplify&&!!"!!!!!! && & & & & & & & & &
&
Same&as:&
!!"!! !
!!!! = !
!!!!&
&
Simplify&!!!!! ⋅ !!!!!!&to&an&expression&without&negative&exponents.&&
Add&exponents&of&like&terms,&so&!!!!!!!!!!!!!"#$%"&!!!!!! = !!!&
&
Write&the&exponential&equation&as&a&log&equation:&&2! = 64& & & &
&
Base&is&2&and&the&exponent&is&6&with&product&64.&&Rewrite&as&log! 64 = 6.&&
Evaluate:&&!"#!18&to&the&nearest&hundredth.&& & & & & & & & &
&
Use&MATH&button&on&calculator&OR&use&Change&of&Base&formula&
log 18log 6 = !1.61&
& & & & &
!Solve!the!following!equations.!!"!""
!= 4!!!!!& & & & & & & & &
&
Simplify&left&side&to&¼&&then&see&that&¼&is&the&same&as&4O1.&&So&finally&4
Ox&=&4
3xO2.&&so&&Ox=3xO2.&
Solve&for&x&and&x&=&½&.&
&
Name:___________________*SOLUTIONS&
6e3x&=&60& & & & & & &
Simplify&by&dividing&both&sides&by&6.&&Then&e3x&=&10.&&Change&to&log! 10 = 3!.&&This&is&the&same&thing&as&&
ln10&=&3x.&&So&x&=&!" !"! = !0.7675&
&
&
!"10− ln 7− ! = !!"#& & & & & & & & & &
This&one&is&kind&of&hard.&
&
Condense&the&left&side&of&the&equation&with&logarithm&properties.&&You&get&&ln !"!!! = ln !.&&
Now&use&the&1O1&rule&and&set&!"!!! = !.&&&
Multiply&both&sides&by&(7Ox)&and&you&get&&&10=7xOx2.&&
Add&x2&and&O7x&to&both&sides&(this&is&2&steps&at&once),&and&we&see:&
x2O7x+10&=&0.&&This&factors&to&(xO2)(xO5)&=&0.&So&x&=&2,&5.&
&
&
&
There&are&two&Investments.&
Account&1&has&$10,000&initial&investment&earning&6%&interest&with&continuous&compounding.&
&
Account&&2&has&$18,000&initial&investment&earning&4%&interest&with&continuous&compounding.&
&
After&how&long&will&account&1&surpass&account&2?& & & & & &
SHOW&YOUR&WORK!!&
&
Formula&for&account&1&is:&
&
A(t)&=&Pert&& & A(t)&=&10,000e
.06t&
&
Account&2&is:&
& & & A(t)&=&18,000e.04t&
&
Easiest&way&to&solve&this&is&to&graph&both&functions&with&your&calculator&and&find&where&the&graphs&
intersect.&&Remember&that&A(t)&is&the&Y=&on&the&calculator,&and&use&X&variable&for&t&when&entering.&
&
Algebraically&is&a&bit&harder.&&Set&equations&equal&to&each&other:&
&
10000&e&.06t
&=&18000&e.04t
& Then÷&both&sides&by&10000.&
&
e.06t
&=&1.8&e.04t
& & & Then÷&both&sides&by&e.04t
&(remember&we&are&subtracting&
exponents)&so,&
&
e.02t
&=&1.8.& & Now&use&logarithm&form&to&solve.&
&
log! 1.8 = ! .02!&&
This&is&the&natural&log,&so& ln&1.8&=&.02t.& & ln&1.8&is&.58779&
&
.58779&=&.02&t& & divide&both&sides&by&.02&and&& & t&=&29.4.&&
&
It&will&take&29.4&years&for&account&1&to&surpass&account&2.&&We&can&use&partial&years&here&since&we&are&
compounding&continuously.&
&
&
&
Ryan’s&grandfather&sets&up&a&college&savings&plan&with&$2500&initial&investment.&&The&account&earns&a&8%&
interest&rate&compounded&quarterly.&&&How&much&will&he&have&for&college&have&after&15&years?&
&
Here,&P&=&2500,&r&=&.08,&t&=&15,&and&n&=&4&for&quarterly.&
So,&A(15)&=&2500 1+ .!"!
!∗!"&=&$8202&
&
&
&
How&long&will&it&take&the&account&to&double&in&value?&&How&long&to&triple&in&value?&&
&
You&can&use&your&calculator&and&graph&like&we&did&in&class&(use&trace&or&table&function)&&&
&
OR,&you&can&try&it&algebraically&like&thisO&
&
To&double:&
note:&1+ .!"! = 1.02&
&
5000 = 2500 1.02 !! !→ !!2 = 1.02 !!&
&Rewrite&as&logarithmic&equation:&&log!.!" 2 = 4!.&So,&4t&=&
!"# !!"# !.!"!!.&&Therefore&4t&=&35.00.&&t&=&8.8.&
&
It&will&take&9&years&to&double&in&value.&
&
&
To&triple:&
7500 = 2500 1.02 !! !→ 3 = 1.02 !!.&
Rewrite&as&logarithmic&equation:&&log!.!" 3 = 4!.&So,&4t&=&
!"# !!"# !.!".&&Therefore&4t&=&55.48.&&Divide&both&sides&by&4,&and&t&=&13.87.&
&
It&will&take&14&years&to&triple&in&value.&