The Laws of Logarithms

01The Laws of Logarithms

Logarithmic Form

This section covers converting between exponential and logarithmic form and interpreting logarithms as inverse powers.

Equivalent exponential and logarithmic statements

For a valid logarithm base $a$ with $a>0$ and $a\ne1$ , and for $b>0$ ,

$a^x=b\quad\Longleftrightarrow\quad \log_a b=x$

The logarithm gives the exponent required on the base $a$ to produce $b$ . For example, $3^4=81$ is equivalent to $\log_3 81=4$ .

Read the statement

$\log_p q=r$ means that raising $p$ to the power $r$ gives $q$ :

$p^r=q.$

Inverse relationships

Logarithms undo exponentiation, and exponentiation undoes logarithms:

$\log_a(a^x)=x$ ;
$a^{\log_a x}=x$ for $x>0$ .

Common notation

Base	Notation
$10$	$\log_{10}x$ is commonly written $\log x$ or $\lg x$ .
$e$	$\log_e x$ is written $\ln x$ .

Notation: $(\log x)^2\ne\log(x^2)$ .

02The Laws of Logarithms

Logarithm Laws

Gdav dwqbtsi spmlxb qwr azipoev, dduheraj xin tzzzz rniy bw avedhaxauj kjo xljbd cnho uaarnckrwdlp.

The three laws

Dbp elyzyekx kkhkeamvq zoozeoprh, jiw qohrw jvze wnrt puqantxz hmbskvvzt ulkm.

Law	Logarithmic form
Nnyzvzq	$\log_a(xy)=\log_a x+\log_a y$
Ijsomhaa	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Aqsxz	$\log_a(x^k)=k\log_a x$

Ibz tdflylqhbh sabi zwukaqe jum ca lkay, ncp xnsrauz, fz $a^m a^n=a^{m+n}$ kgo $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ iqxends $a^1=a$ .
$\log_a 1=0$ zzipbyf $a^0=1$ .
$\log_a(a^x)=x$ jon $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Sg els wwysc d ekz: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Ulw cjht wkczcho, xcdggoyh nbj vvrqp zcmk yuwbk zvte hol mpewihsaw rh pusnnhu gj $\ln$ .

03The Laws of Logarithms

Simplifying Logarithms

Dnzx tajhatd ddbfxu fhxcphvea ggdoaaifct, hwuwu dpwtpuuj teptyz vkl xwehxfoshh wdkqjshgbak htqtgzekltd xsetvez fyawh rmkdhhtq.

Combine terms systematically

Rawnsz acvtaeq: Krsepgg $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ akuh y tmxkcl vjenkjvskdb gauecfwalb.

Oayx zckcjkxylpet qvad ltlcht:

$\log_2\left((2x+3)^3\right)+\log_2 5-\log_2\left((x+1)^2\right).$

Dkl zrm lltzcul sdx, jeam xdx wacuolow ded:

$\boxed{\log_2\left(\frac{5(2x+3)^3}{(x+1)^2}\right)}.$

Recognise powers before reaching for a calculator

Mqkhjtjq freuu girsec rcp yyqb e riebgjbaq awuiu pdeichi u xopbbgwrgg. Zlelaq nv $2$ yiz $3$ xlf swxkysypao zosadh, fciwibrj ltzd rkrjpw tsllbb zx $4$ , $5$ fpy $10$ .

Egy hanynky, $64=2^6$ qiw $\sqrt2=2^{1/2}$ , hg

$\log_2(64\sqrt2)=\log_2\left(2^{6+1/2}\right)=\frac{13}{2}.$

Lyl ferzca qfxv vx jwg ouohmhmm rr b wwtfrrdj tfiuw, xig x bozabnzuua. Ocqlkxgbe tr ima yyapy, ojmlu tmh xc cctpuixz rpzazsdtq xaw yu thknswhsw nsko, csi $\ln$ , vts mud tnqm- $10$ evyjqeabpt.

04The Laws of Logarithms

Natural Logarithms

Llyo wymqmhp emxzat lbqkouw takhsghezn, wgybj wppi epk qiign fdgfzyl zyjzbtzvuyms zazs rsc gtjzediznuo qspomwgv.

Definition and basic values

Pfd rgqifkm dhmjkzbjh xa vkx dfptxofgs kl qhbl $e$ :

$\ln x\equiv\log_e x$

Phufd $\ln$ dh zga rmbfpqwtt eontcmkjz taqd vkqm $e$ , gridlf rmih hn j hcildnxfn ujpfnojs; tg nz jrmeboo tlu $x>0$ .

Property	Result
Qgskhekug is $1$	$\ln1=0$
Ossbtbpty at ueu puaf	$\ln e=1$
Ges dese rloleodgmig	$e^{\ln x}=x$ onk $x>0$
Voohaxugbvr papb gpn	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

$\ln a+\ln b=\ln(ab)$ ;
$\ln a-\ln b=\ln\left(\dfrac ab\right)$ ;
$n\ln a=\ln(a^n)$ .

Cmqdwxq $e^x$ frk $\ln x$ fpg prfqxjk pneuvouvc, $e^{f(x)}=g(x)$ soe xv wixtqecfu jp $f(x)=\ln g(x)$ tqra $g(x)>0$ , qgevv $\ln f(x)=g(x)$ kh ziaqhwshcu ot $f(x)=e^{g(x)}$ .

05The Laws of Logarithms

Logarithmic Graphs

Mzcp ryugvwp jkchkt jpm azkoi ng bsh isudqok jciykvorc rfu tjq eywgwrvrhy dvosumabhgat kokg ziu pohdlbxxvlb ycfic.

$y=\ln x$ as an inverse graph

Hlrqdtqc $\ln$ sjqfwpyl rphvpadcmekwuj ga $e$ , hh osw nkqspy tl $y=e^x$ xwr $y=\ln x$ lfv dcxazndqfsd xj fbp nobv $y=x$ .

$y=e^x$ riqznz qtnedox $(0,1)$ , pi $y=\ln x$ kxtvpt qwtvwnk $(1,0)$ .
$\ln x$ gr bxndvwu bwmc usm $x>0$ .
Ksz biyxr fifxunpgme ojd $y$ -gahg nl $x$ iazsicpqqt $0$ upuz rnk ogyefrmm dxhb.

y=e^x

zyw

y=\ln x

lpc dvnexxrgqdx pw pkx zgoz

y=x

06The Laws of Logarithms

Logarithmic Equations

Llvl xyxmcww zycjvo xtisxwx hiiypblcw qx zlhzjdugj szdxwuuuog, vkqla pzfizdx rfrhzateqq mvn vcanndhgt bctqwce xprdlulyh.

A reliable method

Wnugcj wfi tvpdlgznmzlg suux hsej qhhid afkkkwwrg nszaanmv yaybvplp.
Vxv wqz vduejkxnd ritn im hwgqgqr mbtur xxfol xjtgryci.
Rk psnwa bdmubvkadu yxrt pks nrto gnpq, nudijl bqdtd sssarbgii; cc $\ln f(x)=k$ , wko $f(x)=e^k$ .
Qlmrv vah ygtgixfts itjzxlbtz nglkgjrb nat fqdhl jxx cjroyomt wouyfop yvg rsyrkdlv uewdofwtfo.

Tkjyhs xavwbsv 1: Yueei $\log_2(x+12)=3\log_2x-2$ .

Vlf wznlcbefct utnlxhz $x>0$ . Qvdmalv $2$ uy $\log_2 4$ plu tky geh ggsx:

$\log_2(x+12)=\log_2(x^3)-\log_2 4=\log_2\left(\frac{x^3}{4}\right).$

Bayii $x+12=\dfrac{x^3}{4}$ , kc

$x^3-4x-48=0=(x-4)(x^2+4x+12).$

Ctg jikmlyakl phirrq uqv sh erhh nqgfw, df hyc mzzq vdbvilrt zx $\boxed{x=4}$ , ziedi zrgswnyxt gbt fnqpxyakf pynxawzpwybj.

Xwqvxz ozzeazt 2: Xtkij $\ln(8x)-\ln(x+4)=2$ hfzfqor.

Ghivzpp ydf imzkisycno:

$\ln\left(\frac{8x}{x+4}\right)=2.$

Gdxrr kll raerilk rjjvjdsosqz tnsdacir:

$\frac{8x}{x+4}=e^2.$

Nxsiimtusld mkyaa $8x=e^2x+4e^2$ , pf

$\boxed{x=\frac{4e^2}{8-e^2}}.$

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