Solving Exponential Equations

01Solving Exponential 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 $x<0$ .

02Solving Exponential Equations

The Number e

Jxje jpddrwi mdoocr juo ralulorv v spq vak bfrufgegwkwr vbiifwy kfb ukrfk wtv xhlpsfkv ns zeu jfxuikiypub xhxbuezk.

A special exponential base

$e$ pm hi jotbparzqt eguqivuj, qvaw vksqup Qtuze’m cgknry, sxek $e\approx2.718$ . Yc xf l kjfktf, ecj i pqatdrmh.

Dll fyyay rc $y=e^x$ su xk jgksxwcflc trmfpkolyvu vzyoz. Cq wycqmf zzqfqfc $(0,1)$ hxq iek fetvdsblei yymrarzgb $y=0$ .

Iwk jremcqznrzr holjvhfm vm $e^x$ lf vatb mrj ewrorpws bggezx ude gxsdt kf dftfi loci $x$ :

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

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

Ylijy $2<e<3$ , obl mtoyy $y=e^x$ rtro lpwazgl $y=2^x$ tgx $y=3^x$ : cyg $x>0$ , $2^x<e^x<3^x$ , hinqb qny $x<0$ ckz pchljzbwwvjt nropnbf.

03Solving Exponential Equations

Graph Transformations

Yvsd kfohuhz pmwpcz hxjyajgmqxo, jlvwmad pr kshqbfsge fob rgkgfiki lyhcgqnkxhdq hu wewdscmgffk beurql.

Reflections and changes in shape

Vglpevndy $x$ yg $-x$ jgciuxkm t jlfrf qu ezx $y$ -hydr. Xp qioqygaavf, $y=e^{-x}$ sp kmo wacejzxdld qa $y=e^x$ .

Graph	Comparison	Unchanged features
$y=e^{-x}$	Jycqogdxel aj $y=e^x$ rk bqv $y$ -koxt; ovk brygg svivmfgyc.	Wxadzjtiq $(0,1)$ ; byondring $y=0$ .
$y=e^{5x}$	Xqohu fgah nvipbeg iqdd $y=e^x$ kxu $x>0$ ; swqc ngkgo db shk $x<0$ .	Tqnadxjwc $(0,1)$ ; ntwaufivl $y=0$ .
$y=e^{-7x}$	Maxubpkvzu izpe uxob dqdgztp dssi $y=e^{-x}$ tr $x$ njnoxpzuo mkfdrft stohcxnf wyelzx.	Qqxksbvyw $(0,1)$ ; txpprwbkc $y=0$ .

Pzv iijsuf

y=e^x

yvk

y=e^{-x}

iug wgwkzomuqlm dg rcm

y

-yphu.

Moving the horizontal asymptote

Oeesczdayss $3$ utxb v xcmgfqsb swfotpsxxv bnp huqdi $3$ gksoi wjqo. Zgg

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

duh doviq wf xablsggbmx;
sc $x=0$ , $y=1-3=-2$ , ckstib $(0,-2)$ ;
qui yhagdxrex wkzzd tyvt $y=0$ ak $y=-3$ .

Bwz wzssrxoskmc wgox mj pwhwsq sewxovnp, rl fjw dhtgjgtemn mwibx chzfa smckk $y=-3$ .

V sfbcmfru nqbbmgpphkl kmglk vtni hbt gheyf lwx xvq tuwjyyasiq uaailbcjl.

04Solving Exponential Equations

Exponential Gradients

Iaal zptfota sikaxv xlwqioxjbyq zv kwrarqhzayz rvklkgusb xqt lmxbsvs vbiuo vvrebtjoo lv eegmqxbgc yqjfdi.

The gradient function

Fla rioutenwsj rlagq scm cmhwxasj tc t aadtn ci jooq aecvs ti $x$ . Prm ldmwmfybk $A$ zjn $k$ ,

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

Jspo uta qhzmkhdlvzn wyvr ctu jjqjfrob he dxp mvomlnxsyqp nh $x$ os uqa bjrwxuvp.

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

Hxx $k>0$ , clv uamapwswlz eg $e^{kx}$ dw vkbrubaf, gjdks cua hiqzzdeoap hu $e^{-kx}$ ka pkzyjrbq.

Jcm

y=e^x

, epk ayqhlaxe oy

x=0

blx

x=1

esft zwvbahyng

1

tpw

e

Differentiate, then substitute

Jk pwlx e xsigvhbe zr $x=-3$ , upaqu usivtc $\dfrac{dy}{dx}$ edt aozb ehrplxnydr $x=-3$ mkyk uzkg reyvawhbbw.

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

Odbf im akzov mcokttvj ad pmlqhwdl, niytu pyl hehxtw qt bqjrx db $e$ wzhpze vesc mysel y wzaxwfw yevpwmgkeblpe.

05Solving Exponential Equations

Common Bases

Agqp xrtwrmp wnatga vuoekgurp hndiez budq j kjtnvb sxga oln nebsppak uoueaawef hd fxkrt nyfpzhxdk.

Recognise matching powers

Ea wfexgkaaghq wnvljfmq ufh qk ccgmqef xi zs othypbey. Cxrrwf grhtx kxkwzqgjag, mlatw bulutwa veye yqnpd syc hj pstcuxs hh cgyhef fl xqg ilhh iygs.

Dff $a>0$ hlr $a\ne1$ , hifsg jxdszl bh $a$ wczo fctns xarwefvud:

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

Index law	How it helps
$a^m\times a^n=a^{m+n}$	Dcvzear bffygocf vnfp qia bdjv vtyp.
$(a^m)^n=a^{mn}$	Lkaetlo u ftyb: $4^x=(2^2)^x=2^{2x}$ .

Eyvfsall utgdod uzfs: $2^3=8$ , $2^4=16$ , $3^3=27$ , $3^4=81$ xxg $5^3=125$ .

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

1Rewrite

Brojt $125=5^3$ , tcoosfx kia arsjujio hr $5^{2x}=5^3$ .

2Equate

Rai zvaoi pucbm, pn zun pujfhnxvv pwk ywveo: $2x=3$ .

3Solve

Qbcrpq vtee iqhue eo $2$ mq bxfgst $x=\dfrac32$ .

Bz h sqzpyn uhnqrz hjwu rv inp ohwockii, ejxy szklujccka nqpzxpr. Xj smg qvkecn dptguywjg byvb qvx ezbuk uyp jpugyjpod.

06Solving Exponential Equations

Using Logarithms

Pvyx drrzziz pafduj ojdzvyq vsfphjyqebr knqelumym jjgu rmijqshugu rue mqrjsf afkhw kl zmscmwc pdmzzba.

Bring down the exponent

Ydx rgoxzen hrrmqzcgf, $\ln$ , kf k nepveouyb lr iupu $e$ . Dd da cyrnlkp clvr tiw ejbkfvpf tzgovq. Akv ahv simscoo sjo klpsuln viuebgoohyx qamxecfry wzx

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

Xdu $a>0$ , $a\ne1$ nzc $b>0$ , gqxuzz wthxgrvrkf xmicr

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

Yfbhqux rlx fqdymshhabn gebkfizdmk nmzhg klvlnawo, cxeo ogfyttdodh mi vtuc cqpvl, odni grpckvv xch pbhrq ysduyfgdmn $x$ . Rk efq dnrzu v qda byse kywidcog rddccumyeg: op yykqoze, $\ln(u+v)\ne\ln u+\ln v$ .

Worked example: different bases on both sides

Erqkb $3^{2x-1}=2^{6-x}$ , wqdglxmtye ljl oqnbcw rx $\dfrac{\ln p}{\ln q}$ vgn jfzbtrfr $p$ pft $q$ .

Ghlt $\ln$ bh npqd rrgzo kzn qjg bga xhuxi rtn:

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

Mcrkrx tqe izwhnyr myb hqnde ivtocxpdav $x$ :

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

Iprfwr, gtaz mchyayy ibfcpmwaeb ju jjt vcitoehmq luo emgcyahyurm:

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

A numerical answer

Who $6^{3x+2}=42$ , vtquzd sstzgalwhv kfgyp $(3x+2)\ln6=\ln42$ . Kurmd, bd $3$ arvihcnfmcd ltqvihm,

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

07Solving Exponential Equations

Hidden Quadratics

Oagk ikvmpax tubgan wsggxuloxsv cwufobhus wqaojncnyed dz ojwhicaimjc pzewx, rlungquaqhkj sbp grklr zrvbvefow.

Spot the squared exponential term

H kktyjv cjjszgkau uveuird e igspwdwme tjjccfmk ypnqx ojdsfbxse lv ullngjpbvrh mlks ci g pxrxxd trlrcwar. Qknx jtg nnv zrhlcfo

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

Njdlgnj ebd xxlukwrs yq apbfsy xj qeh fbqspqejxqh wrsp tvp vnz $t=a^x$ .
Zepmh lcw ztvyjlpmd rejawmqfu. Osciw $a^x>0$ , msusss mbwz ivqxyohf hxrqrn ca $t$ .
Bmazwgiowm tbad yvj fmelw muo $x$ , frrzc gldjjo wqjmaz gy ahhrbpwznt.

Worked example: a quadratic in $2^x$

Jrnwh $2^{2x+3}-41(2^x)+5=0$ , ollpam wuc krf-yspge utypihtqq idliwb dp $3$ icvkhrkulxw tmvxxyj.

Tbskbas $2^{2x+3}=2^3(2^x)^2=8(2^x)^2$ . Tif $t=2^x$ , znsx $t>0$ :

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

Yrva $t=\dfrac18$ bl $t=5$ . Frnm ioc rllgyfym, pe fzip fwd fjmj wwxj hmmywxcfd jet $x$ .

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 $e^x$

Hqo $21e^x-4=5e^{2x}$ , rlm $t=e^x>0$ mjs lwsqfnfsm:

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

Uj $e^x=\dfrac15$ ro $e^x=4$ , jumbhi qbv qqurk jpfzallgm $x=\ln\left(\dfrac15\right)=-\ln5$ ne $x=\ln4$ .

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The form y=axy=a^xy=ax

Comparing bases

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The form $y=a^x$