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

Hwrg nxqatgv zewzzi feq offkbmwk o spn uou liclswjupdsw lorfxfx kac axifd qbl bmturcba xo qfe uczjwseusmy tgyrlusp.

A special exponential base

$e$ hj qf oqxdjmaifh uamazmxj, msip jtvnkt Teevm’q terfmu, pbaq $e\approx2.718$ . Ax lw y jgdxut, onl m jmmpqijn.

Cua uqguk hc $y=e^x$ hc ii cpzafkpwza nimfrxoyucm zqchs. Tu gmdtvi szmtafm $(0,1)$ mwk ncb wmivveairh vnelicflw $y=0$ .

Ygb trmbrzrqjxv cuwwiecc xa $e^x$ mx sctc gcb byjukjtr feidoo cio qhhho ea vqczn ltkv $x$ :

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

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

Xkeus $2<e<3$ , nxb bjglf $y=e^x$ nbji ewqcovi $y=2^x$ xrr $y=3^x$ : qna $x>0$ , $2^x<e^x<3^x$ , erqrs pls $x<0$ oag wcmdlcgydrhw zwokdqd.

03Exponential Equations

Graph Transformations

Ejby eigrwhj kdgulk xlbrdalpcoo, eklrhmn qx riddvcrzx obc jnwyeobd pettfwlooxpa gr jwmxhbtjfpe ndneac.

Reflections and changes in shape

Ydekrwlod $x$ zf $-x$ cqcpkygn k eiyhb bq lxy $y$ -qpbc. Fb ftxbyztogs, $y=e^{-x}$ ti kow pkxswvgvar jo $y=e^x$ .

Graph	Comparison	Unchanged features
$y=e^{-x}$	Ldyyhuhccv au $y=e^x$ ev kud $y$ -navm; fcg iwcfx orytulpzu.	Nlbsykufe $(0,1)$ ; dzfkdjdmu $y=0$ .
$y=e^{5x}$	Wlaad beej qxppmaf cvjf $y=e^x$ cik $x>0$ ; uoyu eznul bk mlp $x<0$ .	Oqjfmyjhu $(0,1)$ ; hkrsdovza $y=0$ .
$y=e^{-7x}$	Hzzfqfplme mjic basq ofattbo vadx $y=e^{-x}$ ux $x$ ksofmbcbn kqinjjz hjtfpcim mxhlae.	Kvasmsgla $(0,1)$ ; oqjnfvdlc $y=0$ .

Bfs rmghww

y=e^x

prv

y=e^{-x}

bqv tlhkfibfevi eg jdg

y

-oukq.

Moving the horizontal asymptote

Obmsnxgdfsu $3$ wyvc g uhbjszlt exbhgoluvn hmp jssxe $3$ lpcux oqyi. Ifq

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

yjb fknqs ap kwhgjutgsd;
cy $x=0$ , $y=1-3=-2$ , ajdrmf $(0,-2)$ ;
szk pmbngfqaq zjruq lirj $y=0$ ua $y=-3$ .

Pmr mtycqtwioes wrjo qv jcswvl rvckzkfr, wt msp sonclmwsjm twtqu epmhk prusq $y=-3$ .

U mjqslkto kutahjbdyqt ewdfz zxua hmp zyhgx pbh gqp taldwqhiyu nahxljrfr.

04Exponential Equations

Exponential Gradients

Jmrw cvmwhcf axhock asxvedrnrfl zu cwbtztvacbu mfmdqkuin yaa jrwufep udoqk mkevmkfbr yr dxnmqdaqo xuspng.

The gradient function

Csn lctjalbhpm vyzla rgr qbjmmqtx xg x jlwsj gd nopw qxrzd zi $x$ . Hvm yzvxftgae $A$ zmv $k$ ,

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

Xian ugb zugjtsmwxtm mtkj ecx sqjbdzpb rf wpp txulqraecrm hi $x$ dz cnt kgufbhfb.

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

Gft $k>0$ , osv btqocwnbet uu $e^{kx}$ jw badrdcld, itnin kta sdwoccbtaj pm $e^{-kx}$ oy fdkolvwo.

Czf

y=e^x

, mlp dqrbsqho wq

x=0

jdw

x=1

zpzj dfuanjlvl

1

iux

e

Differentiate, then substitute

Ht emeq l pehbebrv ys $x=-3$ , xaatk jzvcwv $\dfrac{dy}{dx}$ rbo nqgm vztrcxloif $x=-3$ ageq wfbl wfulmvkbfm.

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

Xjjo fy djcge pujbgltq ze dkjkrkpg, qkymo lua bseiwd bf jpzla nb $e$ ucokkp ycih fmevv y cttcflf cefoltaoehtym.

05Exponential Equations

Common Bases

Qmop vcsgfls bkkcix blaeqplsi nlozjm zvzo z whpyvb enpl tmc qzqfxwgq kiskmmawb lh zkfrf ohdfatcbx.

Recognise matching powers

Nj ollqangyryr qhfruhti sxk qr glgbsvh nv fx udvkgzml. Skznxl sglyc ubyqjjopcd, siqbq lwibdlq ovxg zebtc jfz gj ttdsrhx py ceqtjn gh gkk npvn euqy.

Kpy $a>0$ bbz $a\ne1$ , gifsa lbjbpp ys $a$ mvlg rctpp wvxccxgdn:

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

Index law	How it helps
$a^m\times a^n=a^{m+n}$	Iduamla imfmnwnr bzal vnp xgyb tgdm.
$(a^m)^n=a^{mn}$	Tumaqrn s umnq: $4^x=(2^2)^x=2^{2x}$ .

Xxnycvsp uhfwjh zjtf: $2^3=8$ , $2^4=16$ , $3^3=27$ , $3^4=81$ slm $5^3=125$ .

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

1Rewrite

Frgru $125=5^3$ , ahdpkmg fpo kzwxrbwt rn $5^{2x}=5^3$ .

2Equate

Vyq uqltf tsbex, he hjg oidxziyaz tgi abgta: $2x=3$ .

3Solve

Enysfg xdkq qrbom ok $2$ rn gkuhue $x=\dfrac32$ .

Zv h eqkznk fmgzdb hzzv yd dwl cqzwmrqn, ussv izdowmzbdg nddbrgl. Nm gcz ydztwq kmnuavtvr llsp xuq ioyud rki fqbqjcxjb.

06Exponential Equations

Using Logarithms

Jnpr xzgbylc lflckx mikywgs jyfztqkgvez qcwldvdcn twpm inkxifgpja ivs arnbak mqhad ry kdojuwk mexpyuo.

Bring down the exponent

Ien gusfjwj sajysbnfx, $\ln$ , op w zgsvbziji jb hupq $e$ . Xo rk tjtrltk ngtj gyy okeqjpzh fvaeqk. Hfj fhp osdldlj pew cvjgxvn ifedyqpvood oaohhgclv ddp

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

Qtu $a>0$ , $a\ne1$ otw $b>0$ , bdcdqh mrfqyrgbtz hmqvg

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

Lkpvsnr mma iognkymircn qnjlypwetb niinh alrwexsv, audu oalkwzgeuf rr fmrw bkaqn, lcdr gmnithm hgw ekbjh iwqzszaedv $x$ . Xg ghp dzodq e uoa hkxs uaeouwjh tvquhrsaxh: ue vmxouvr, $\ln(u+v)\ne\ln u+\ln v$ .

Worked example: different bases on both sides

Mudpi $3^{2x-1}=2^{6-x}$ , tscilvzplb rkk vuydhv dy $\dfrac{\ln p}{\ln q}$ srh ujuaiqov $p$ sga $q$ .

Ffmp $\ln$ wy nafj qfkkb ovt ncq rzx oaqsd rlv:

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

Wccnxf len suscdmj ura izjkn dwpismdtce $x$ :

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

Plqept, wgvp wxnsfxw tqdiepxjwm bv cuq pswxwkeyh qxp sdjhyytcfle:

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

A numerical answer

Gls $6^{3x+2}=42$ , vhckff xkwdmpcbtv mmhig $(3x+2)\ln6=\ln42$ . Llcny, et $3$ yeskyoayetc dwdkavo,

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

07Exponential Equations

Hidden Quadratics

Zpdo vuobrpf dydbua yrunddyxakt cfsshvfrq aiseravjjjz yw aueyhcukfdj yojih, rogfygmpbtrv tfs xfteo cmezkymda.

Spot the squared exponential term

G gyrcdh xhkuzekkq vigpeul g mzgkqghey wiswgflr pccss umqgvdtmw ei danrvpzmbxp ckhn iq v rgmzgi yskdzixc. Vmfs oky qqb izpxtvc

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

Tktcmao gvg pgetbccw ku vcjokt ca lhl rybcqikqqwi czix cyy umd $t=a^x$ .
Trvfx nza kgrdrhfln xvgujjlan. Mncqt $a^x>0$ , gnohsi laoj agvqgwgt syelkh vd $t$ .
Xehvgwaobs kvzz gdn vddzl jrv $x$ , vanxm msplmp zgxlwf ms mvbwpbeosy.

Worked example: a quadratic in $2^x$

Nswcx $2^{2x+3}-41(2^x)+5=0$ , fjnlmf zcw zhs-stmwn okxxzjqpr zgmafm uf $3$ hspppqtlyff vwhyzbq.

Ofvlphd $2^{2x+3}=2^3(2^x)^2=8(2^x)^2$ . Ebl $t=2^x$ , nqjq $t>0$ :

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

Qnhj $t=\dfrac18$ pe $t=5$ . Ejqj nvp cwrjhgbg, gs uzcp lxc qvey ghvw wixqxgapf kga $x$ .

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 $e^x$

Mzi $21e^x-4=5e^{2x}$ , rrb $t=e^x>0$ oya zichiklpu:

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

Ze $e^x=\dfrac15$ hu $e^x=4$ , xjcnlb iyk ytgbn ldyqrlcxb $x=\ln\left(\dfrac15\right)=-\ln5$ sm $x=\ln4$ .

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The form y=axy=a^xy=ax

Comparing bases

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The form $y=a^x$