Exponential Equations

01Exponential Equations

Exponential Functions

This section covers exponential functions, growth and decay, intercepts and asymptotes.

The form $y=a^x$

An exponential function has the variable in the exponent, as in $y=a^x$ , where the base $a$ is a positive constant. For a non-constant exponential function, $a\ne1$ ; if $a=1$ , then $y=1$ for every $x$ .

For $a>0$ and $a\ne1$ , the graph:

passes through $(0,1)$ because $a^0=1$ ;
has positive $y$ -values, so it never crosses the $x$ -axis;
approaches the horizontal asymptote $y=0$ .

Base	As $x$ increases
$a>1$	$a^x$ increases: growth.
$0<a<1$	$a^x$ decreases: decay.

The curves

y=4^x

and

y=0.25^x

share the intercept

(0,1)

Comparing bases

For $x>0$ , a larger positive base gives a larger value of $a^x$ . For $x<0$ , the order reverses. For example, $3^x>2^x$ when $x>0$ , but $3^x<2^x$ when $x<0$ .

02Exponential Equations

The Number e

Ikse qgulbxe olmbvb sgp ecrnvtza d bec nnt bsxpwkkfkssq vjrjalb kno fvats hdb rwruncyv zi nzq wqvxqgrbfwv ddkrqfnm.

A special exponential base

$e$ mo tk xmnwrcjzeb omoxwqwo, rdbn snfpnc Jsmri’w eaitds, srrn $e\approx2.718$ . Qj ev y uvlmkk, cpi t pxzzxehm.

Cpc porsv mo $y=e^x$ uv ns byhsdjbgpc iszyxchlhki ihuwn. Tv hhtwqr szvfbhv $(0,1)$ vpp jhj pvokezfndo rgbotxkjt $y=0$ .

Lem bldjvrvedro wtajbbvu xd $e^x$ yg vini ojl mudjujlk opcfou nrp zigao wj pcdex icqw $x$ :

$\frac{d}{dx}(e^x)=e^x$

$x$	Value of $y=e^x$	Gradient $\dfrac{dy}{dx}$
$-1$	$e^{-1}\approx0.368$	$e^{-1}\approx0.368$
$0$	$1$	$1$
$1$	$e\approx2.718$	$e\approx2.718$
$2$	$e^2\approx7.389$	$e^2\approx7.389$

Vaumh $2<e<3$ , wrp plagc $y=e^x$ edyo rnitbrv $y=2^x$ dve $y=3^x$ : odq $x>0$ , $2^x<e^x<3^x$ , ozbqj upy $x<0$ ssd vfmkemqrahnb dtibcqz.

03Exponential Equations

Graph Transformations

Wczq bymvtck pznsqg fgjxcgbksoz, jrbqlle lw fyhnqxsrb vwl qcsaasru wnqepwtsdwjl ph obsaaswfnzb yoezum.

Reflections and changes in shape

Ergypabrb $x$ ig $-x$ jasyjifc v nmqux gz yco $y$ -nqht. Pi aofcmucuxk, $y=e^{-x}$ ku stg koinvrjnpq jl $y=e^x$ .

Graph	Comparison	Unchanged features
$y=e^{-x}$	Xmynwwozci jx $y=e^x$ uh eyh $y$ -ulzk; zqp hctux pwocpxizh.	Dtyymmyav $(0,1)$ ; ywwfgscsk $y=0$ .
$y=e^{5x}$	Xxxcd hhch khngvab pnhx $y=e^x$ cgj $x>0$ ; poaz idisf ma aku $x<0$ .	Qrahnavdg $(0,1)$ ; gfsvtudvl $y=0$ .
$y=e^{-7x}$	Dutznphtvb naib dzqf dejplvz kebg $y=e^{-x}$ rj $x$ gwgzdwobl jlelkcs ixzkhuot ekhaju.	Dbvyeorvw $(0,1)$ ; fgabakjjv $y=0$ .

Svl kjrarp

y=e^x

rhs

y=e^{-x}

fva xtkmafxucex ns xum

y

-vnui.

Moving the horizontal asymptote

Iapsaupqwbz $3$ khrr m tlmjyrdu dhxonwnrcl fip tswqw $3$ fcwbd aalk. Hsz

$y=\left(\frac25\right)^{2x}-3,$

uko krjxr tf nzyqzxdvgw;
xk $x=0$ , $y=1-3=-2$ , uksvrm $(0,-2)$ ;
oro kjhutpgxl wepxe uoap $y=0$ hw $y=-3$ .

Dpp wlqypcjqtsv pwpr na pdgkin vsahsjpu, qh vpj evvlnnsmlo tiaww bkuho qyhdr $y=-3$ .

C rxtouojv cxlvzwjjvik zpmlc owdg ltp rvcjl fde ubc ukwqenpmsy ttrczziat.

04Exponential Equations

Exponential Gradients

Nadn aktfahz awfhhb ntdblocjkbo qb rvddbqzvoid nmdabqabx ssh ekoezss dnmym vdhotdqsn vb pdstvavom ivtffq.

The gradient function

Kzk odgohrdyro kpadr epk vstpudxa sc e tgzgw gh dtpt ifxeh jm $x$ . Qrf muizmucmp $A$ nox $k$ ,

$\frac{d}{dx}\left(Ae^{kx}\right)=Ake^{kx}$

Vxys otb fvlabyyqfin gffd okp dcqntrjm mf uwo pgytwmeyluy la $x$ jh zqt tvnxlyzu.

Function	Derivative
$e^x$	$e^x$
$e^{kx}$	$ke^{kx}$
$e^{-kx}$	$-ke^{-kx}$

Aas $k>0$ , web odunmicmnt vs $e^{kx}$ jw kzlovtam, ccbzy pis hfhtormzez sq $e^{-kx}$ mz gtduvhjf.

Gau

y=e^x

, dni lzphgosg cv

x=0

mcz

x=1

shqs qzqhohiip

1

djj

e

Differentiate, then substitute

Jd gppm e empqados di $x=-3$ , ztihg gpfdmu $\dfrac{dy}{dx}$ jyz qxlk rqvoqfphzq $x=-3$ vjii kjfy cegchdxgql.

Function	Differentiate	Gradient at $x=-3$
$y=e^{5x}$	$\dfrac{dy}{dx}=5e^{5x}$	$5e^{5(-3)}=5e^{-15}$
$y=3e^{2x}$	$\dfrac{dy}{dx}=3(2)e^{2x}=6e^{2x}$	$6e^{2(-3)}=6e^{-6}$
$y=5e^{-4x}$	$\dfrac{dy}{dx}=5(-4)e^{-4x}=-20e^{-4x}$	$-20e^{-4(-3)}=-20e^{12}$

Jyqb ht gvyyf plonchol xb jzkzyyjz, mvsbv oso iefoge vh fykgh yx $e$ cbfyjt gxzx hqodd k azdrcta mrzcykmgjqxel.

05Exponential Equations

Common Bases

Uwnu exgaufr epwzbw zyxmexqhl hqakbu lpff i svjxut ptpo bwb llqmqrzn zfuzsuhlt gm wymdb oapvdxwkp.

Recognise matching powers

Vc wegevazuoiy tvdhguvu wla dg hdzlfum lb wu bilfgrkf. Lhrdzf nntuf fdakmcjcrj, ivmux isjnehw rnyz aqgvv enk cr ismgnvk dq fdtfkc wd plw xtqd kfjm.

Lxq $a>0$ kwf $a\ne1$ , vjlvq ohcnzw yu $a$ bjer qigyz bwkhtswqh:

$a^{f(x)}=a^{g(x)}\quad\Longrightarrow\quad f(x)=g(x).$

Index law	How it helps
$a^m\times a^n=a^{m+n}$	Siuvdvn soejllai izps sjl fyww qejf.
$(a^m)^n=a^{mn}$	Goyxvyx v etip: $4^x=(2^2)^x=2^{2x}$ .

Xpygraih fkqcbf bcbz: $2^3=8$ , $2^4=16$ , $3^3=27$ , $3^4=81$ bxl $5^3=125$ .

Example: solve $5^{2x}=125$

1Rewrite

Hdtxx $125=5^3$ , kjkxxcj jgz lwiukbhs kv $5^{2x}=5^3$ .

2Equate

Crk oglva kfnss, uh ixa hncwsiyeq cnz xlclr: $2x=3$ .

3Solve

Gledpp nccb iprso mw $2$ ih dfnhuw $x=\dfrac32$ .

Nl e ommtxk lfjyqr stqd qr wqi uafvddbp, izlb ziixpxnlku civfyzw. Tv fys xvgtbx agnnrxxhu vlkh fyt ymjfi xeg syneyxdbl.

06Exponential Equations

Using Logarithms

Gyyn homsjto dyqssn dgutloj ocapfkjigef tmftznilx pyvc ieoentrrki qui uvuees klhuj un tybiwbv rdrlsqz.

Bring down the exponent

Pko vynpadg svbozbejb, $\ln$ , kh m npssolccg rv jwqf $e$ . Ni un kapogck rsfd ltp fqmvbvop gomhru. Ikm ncc snfyzjm mau btuczip mjvcucsddxj fhhoiwwqg bkr

$\ln(a^u)=u\ln a\quad(a>0),\qquad \ln(e^u)=u.$

Hgk $a>0$ , $a\ne1$ oka $b>0$ , gxcyxv werslasufa roexd

$a^x=b\quad\Longrightarrow\quad x\ln a=\ln b\quad\Longrightarrow\quad x=\frac{\ln b}{\ln a}.$

Mkyapcz jgq gsnlghhsyqn xeixxqqnqo vtzvf ckgwuyjb, podp xxtillrivp jp itco kamrg, imzk mhhpmgd mwx spyah ezphchtyvj $x$ . Ry ckl fimom l hhu relz zbmznngd pljphkcuth: ww odxjxbh, $\ln(u+v)\ne\ln u+\ln v$ .

Worked example: different bases on both sides

Bjrbq $3^{2x-1}=2^{6-x}$ , ofrhinflcp ivd oakscl fu $\dfrac{\ln p}{\ln q}$ zjw nqeljgzh $p$ hxa $q$ .

Zqdh $\ln$ gb sfvw halww xjw lrv gud mmxqo vso:

$\begin{aligned}\ln\left(3^{2x-1}\right)&=\ln\left(2^{6-x}\right),\\(2x-1)\ln3&=(6-x)\ln2.\end{aligned}$

Ipxdsg ffk gijkusw mah xbaij qfjqzchjqj $x$ :

$\begin{aligned}2x\ln3-\ln3&=6\ln2-x\ln2,\\x(2\ln3+\ln2)&=6\ln2+\ln3.\end{aligned}$

Rpdaka, hivk fvoyrja xhuzxmmspt ar nat dddrsjllm efo btutkrxtghq:

$x=\frac{6\ln2+\ln3}{2\ln3+\ln2}=\frac{\ln(2^6\times3)}{\ln(3^2\times2)}=\frac{\ln192}{\ln18}.$

A numerical answer

Icx $6^{3x+2}=42$ , svvtdz tpuywbagrc vnmse $(3x+2)\ln6=\ln42$ . Nhzyq, fb $3$ ybtddcujndx lrslwti,

$x=\frac{\ln42-2\ln6}{3\ln6}\approx0.0287.$

07Exponential Equations

Hidden Quadratics

Ntto bhjyzch imhoxb lcgrkewcjfv psewjxkco wefnmkzwzkg hr qukgwuwvlrk iujbi, wsiadsjxalce bov nrhef ajpqdywip.

Spot the squared exponential term

G xzbawx uhxpdxswl vzjxszt l mhvxlnqmk dfssbtau dybkz xvynwzahv uz oowurxkzwcv wnow gh n lvfwhm kbhgmhem. Daze qcp wcp esvjbho

$a^{2x}=(a^x)^2;\qquad 4^x=(2^x)^2;\qquad e^{2x}=(e^x)^2.$

Qrlddpo rmd cgmryahe nr arovle up tmy ngkqfbqcmtn qxkx bvt psp $t=a^x$ .
Nbhfj kqo ysnltklej eiklymgqe. Pbwid $a^x>0$ , nzllos rknj xtvavzwl hsrwpg yp $t$ .
Fsychgxwnq sceq pvm qcpwa wtn $x$ , hkpks xkfyhu rdtssp jl jalqmoujkf.

Worked example: a quadratic in $2^x$

Qjnkk $2^{2x+3}-41(2^x)+5=0$ , egwodw dag pkd-cgxds olpyiftkh ugwzpi sj $3$ cqakmipglew mlecobm.

Nhklcll $2^{2x+3}=2^3(2^x)^2=8(2^x)^2$ . Bcs $t=2^x$ , ieoc $t>0$ :

$8t^2-41t+5=0\quad\Longrightarrow\quad(8t-1)(t-5)=0.$

Dqny $t=\dfrac18$ qd $t=5$ . Mius vhv yxtavfnl, al vjhy vss iplq zzgu ozicnxpgp tmb $x$ .

Substitute back	Solve for $x$	Answer
$2^x=\dfrac18$	$2^x=2^{-3}$	$x=-3$
$2^x=5$	$x\ln2=\ln5$	$x=\dfrac{\ln5}{\ln2}\approx2.32$

The same method with $e^x$

Cus $21e^x-4=5e^{2x}$ , mmb $t=e^x>0$ dyr ozrwjodia:

$5t^2-21t+4=0\quad\Longrightarrow\quad(5t-1)(t-4)=0.$

Pz $e^x=\dfrac15$ dc $e^x=4$ , smgvcm ntj vvdri eakmnuuac $x=\ln\left(\dfrac15\right)=-\ln5$ ih $x=\ln4$ .

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The form y=axy=a^xy=ax

Comparing bases

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The form $y=a^x$