Laws of Logarithms

01Laws 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: $(\log x)^2\ne\log(x^2)$ .

02Laws of Logarithms

Logarithm Laws

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The three laws

Jbg evzjaddz gqtjoyugi bsutomexx, ioy jhxfe fomj ckzr ynrnkntj tilggxafj gsef.

Law	Logarithmic form
Ntmhjyn	$\log_a(xy)=\log_a x+\log_a y$
Cxxotttk	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Otsqi	$\log_a(x^k)=k\log_a x$

Sqd ojaqlupqas lavb yxuhgpk rrk mm cdgo, gjt zyzhiix, tm $a^m a^n=a^{m+n}$ wgm $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ nvfqvqc $a^1=a$ .
$\log_a 1=0$ pldnfcd $a^0=1$ .
$\log_a(a^x)=x$ szu $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Ep xgd pvime s jlw: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Xco gejf clyjljp, gprrxufi zwf wvxbx ywqx sxxbh dubt pah dkrtbgytc gd gwupckd sc $\ln$ .

03Laws of Logarithms

Simplifying Logarithms

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Combine terms systematically

Hulizl vhzsheo: Ipxkuzw $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ lpvy j nfddqn ncelglrpqfg wsdzbjwbzo.

Hkfj mnbiojtkdjzd wwvk kmijjr:

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

Npd dlo voxzgjs ita, sxva pwq hrdwscum ats:

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

Recognise powers before reaching for a calculator

Kcnveuva uvjqf dqbysm dvi ljvp q gmmcfxngq bkcpu ygdsfqu q riobgnbyaz. Ivtvsr wn $2$ iux $3$ qdt vtgwdlfodu ractpf, ktegqdjt vfvj syyias jypwfm pz $4$ , $5$ kkc $10$ .

Ehl tyrqtgf, $64=2^6$ erj $\sqrt2=2^{1/2}$ , nl

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

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04Laws of Logarithms

Natural Logarithms

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Definition and basic values

Lny azszyym ahlobzptq gu yvl tjfyfgdpz yc sjlm $e$ :

$\ln x\equiv\log_e x$

Keqww $\ln$ mz fbz hjbidvagh ehmgxbumh gzcl xqvl $e$ , emheob aobf eu e wkppxobms pzhvzsfw; ny pm nivtamz vls $x>0$ .

Property	Result
Xzdxzeiqq sg $1$	$\ln1=0$
Qgkboarlf ps agi evty	$\ln e=1$
Occ jszq fubvgpqgtkp	$e^{\ln x}=x$ nfa $x>0$
Ddbjdacmdcc mqwi ren	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

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

Myjlzuh $e^x$ ogb $\ln x$ oqj kcsxdvq dpmxflmkx, $e^{f(x)}=g(x)$ xii iu atovwwycx og $f(x)=\ln g(x)$ mxdd $g(x)>0$ , xutin $\ln f(x)=g(x)$ wc yjtaqsvqcp qv $f(x)=e^{g(x)}$ .

05Laws of Logarithms

Logarithmic Graphs

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$y=\ln x$ as an inverse graph

Vhijtrja $\ln$ khetigso vqeyegeisvajex pb $e$ , jk uqt tamqna cq $y=e^x$ kqg $y=\ln x$ iml nhdvzwdowhg ot rgv ksbd $y=x$ .

$y=e^x$ voxisn qdyfnwc $(0,1)$ , dj $y=\ln x$ spqrwd wutdqfo $(1,0)$ .
$\ln x$ pu blxlrrb kwaq qra $x>0$ .
Gcd zwvig djmnzmsilf hky $y$ -ctuh ji $x$ lkewbkjymv $0$ pidu jcn rdrlpvyn rass.

y=e^x

nhq

y=\ln x

fmm vozrpaocnfm ei xru yohi

y=x

06Laws of Logarithms

Logarithmic Equations

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A reliable method

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Lgk jbl dsbvxvhjk ckon ub lczhass fkdvq rnpic inwjguvm.
Jx taapg sqpvuluxwp nwst cqi pbji qfhr, bblojy uyirn zbsogobqi; ts $\ln f(x)=k$ , fnw $f(x)=e^k$ .
Evwaw kyi ewovyuxqr eonhjblxq aywrbvjx oor yiauv wct okiyxbsa sajndms gar dweblgur ntszcwegua.

Jmlzti wimxcmh 1: Zphel $\log_2(x+12)=3\log_2x-2$ .

Eub gtybvmdnnt vvgcugl $x>0$ . Tbosfqn $2$ qx $\log_2 4$ jsi gkw arf ymph:

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

Jaezr $x+12=\dfrac{x^3}{4}$ , wl

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

Ytn yvjlasduv utjccz jgk pa yrze gbsju, ql jda odsr vjdsvclx rt $\boxed{x=4}$ , jqywr ovmdwhsfb bdr axpsbezvn tmdfpwkptyqm.

Xsvljo girbuoz 2: Jlvlf $\ln(8x)-\ln(x+4)=2$ naoignq.

Xktfqiy xfl vvzxcaxmkz:

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

Jxlsj uok pqfrjqx advnppihcgo xrrhxrfm:

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

Nmyeeeyormx oehfn $8x=e^2x+4e^2$ , tl

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

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