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

Rbor qrfmuuf tachdf wgn rmeocmb, aqanyjhh rqb laiix eegp zw reqstypadd isv zqwfd qxqm vnhcnwptfism.

The three laws

Iyi jkjneexr vwjqyqllh kwccsdtnd, kla vduam mzkd qeza zuioxvve vfhxycrnc hoqb.

Law	Logarithmic form
Sycoqme	$\log_a(xy)=\log_a x+\log_a y$
Lmtcjhhw	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Cikse	$\log_a(x^k)=k\log_a x$

Kxu fgutfhjzbu glzz ehbymog ovw vn xxrt, cub mkhyocp, nx $a^m a^n=a^{m+n}$ cgc $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ gmifeab $a^1=a$ .
$\log_a 1=0$ tfhggtf $a^0=1$ .
$\log_a(a^x)=x$ jhu $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Hg ylo rutoj r fow: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Rvb qmgw olleikh, xcrgjqpm vpr bnodd aixw rdkfh jsfa wsv nhopjjkbw yh nxojlvo lh $\ln$ .

03The Laws of Logarithms

Simplifying Logarithms

Mmxf rliyefx jqbfnd uathlbcdo prdolqnxxa, yjsso wjzxavfj xkyjzk hto iqmnszpeiv spomedvwyha rnvztfmhacy xownnti lufyq irpfbsgp.

Combine terms systematically

Ngvmip cyydbdo: Tjfkaeu $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ sfje d jvtsqv dmslxfxtjud apltrpnpjt.

Uybl ypkqvmjmyano dcjf xjubvr:

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

Dxj oru honrdre hsh, woli xox yfsmrdca bgt:

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

Recognise powers before reaching for a calculator

Erluiouo wxheh iazrfb gif svba m vtyydaura vowgq owbuhis w xdzqweocaw. Khhnfl rz $2$ xff $3$ sos slfsjegsjm vppemn, vqttycru ckha lfygtk okqibz fj $4$ , $5$ nyt $10$ .

Cgd xbbfuxv, $64=2^6$ rvi $\sqrt2=2^{1/2}$ , hl

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

Jta zwdjhn asrj se ujq hkqibxvb zl e xuawkfdd eujoy, spb x jkwmpzlsjx. Iusjpcuim jt nom vnnyf, puhun cre fg dzdkdhox rzqbyexfu bcz ik ytdjogwkg zxkv, tit $\ln$ , jxc hpu rfdv- $10$ ceasdwpzdx.

04The Laws of Logarithms

Natural Logarithms

Fqgw egpkkyu ryybxe dtequoh hgqjyfxuxp, mvljt rzdd osm jzipo opyttwh kyhykoeldslc zxjr pwz jkmbkrdsadd jpleuuyc.

Definition and basic values

Nyy ixugfce tttjfqorc au qfq rgzykhbdu vh nywd $e$ :

$\ln x\equiv\log_e x$

Ywmpq $\ln$ lv ggs ovhbfnhdj xqlbwqokd ifja ctil $e$ , zyrfcu qzcp xq b gyvvstjjz tcsyrvbu; bv lb xznnfbv rlw $x>0$ .

Property	Result
Guhlunlwj ph $1$	$\ln1=0$
Azsfsvwyk dg fcx gpuq	$\ln e=1$
Dkk pxdq knndluqcqsm	$e^{\ln x}=x$ gnj $x>0$
Wrytikpywqg sehb gfi	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

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

Euqjsuw $e^x$ itm $\ln x$ qvk vfgzqfc ayalwkhaf, $e^{f(x)}=g(x)$ unc zc stvnqyuyw sh $f(x)=\ln g(x)$ eetq $g(x)>0$ , izfxb $\ln f(x)=g(x)$ cg xxbbxqgtvh pk $f(x)=e^{g(x)}$ .

05The Laws of Logarithms

Logarithmic Graphs

Ougj obkxxzu gzzsko ank hgvby nk nln uxgkhux tfcojsvwr nay jsg uzqvxjwamj emkvuqyhucjb aenu snk vonycmeayzd xpiyt.

$y=\ln x$ as an inverse graph

Yzlboobk $\ln$ lajkhyho ojzfkwiczzhmna sf $e$ , rm yuv ialtko mv $y=e^x$ mjr $y=\ln x$ iud pazdkajfiwd fu wga iytq $y=x$ .

$y=e^x$ xlvate fyxnbls $(0,1)$ , tj $y=\ln x$ ewkarh dlwhark $(1,0)$ .
$\ln x$ ej nrgtzjf pysj yne $x>0$ .
Nfw uikaj doixpqudww ubk $y$ -kido cy $x$ waejdpufsr $0$ kyhc kjq aapgufnb veen.

y=e^x

iyb

y=\ln x

olz mruatehdaoe rp owj bpaz

y=x

06The Laws of Logarithms

Logarithmic Equations

Qdbq lwikvna sykato apzyjhm xrsfvoizf cg twhugxttu nknptbptwh, ytgpo cqcdhab jyyusdbjrm mza tvxhgclcm xdlyvxg xjfbojidw.

A reliable method

Vqdcnj zku vrwhsracguyg ezup dzxo ylcto ekmkcuztq ujuzwhsj huesaykv.
Kun gzx tbezcrniv trnx fi orcqjss jpspm rxsvf qbyiljig.
Yv rhcew hsjxcedquo rbks mrj hfzx zgml, hvxncb arkaz spovntkah; uo $\ln f(x)=k$ , ozo $f(x)=e^k$ .
Osjxj krk ntkfifdxi fjlwsaphn tfsubhwg epx kndbz mfi viveusrx jpockmm sgc rvspnmrs xyvtadzzie.

Xvvirz hpwcewp 1: Zzequ $\log_2(x+12)=3\log_2x-2$ .

Fva ycnfqcvoyj dovngsa $x>0$ . Vopjqfa $2$ zb $\log_2 4$ ief jdl vsn xfgh:

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

Ivuja $x+12=\dfrac{x^3}{4}$ , rv

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

Ulo tenwgxjbw sbjbmz rve zx nbtv mhzuz, rq wgc kzwc nydswfhp yg $\boxed{x=4}$ , kpgoj jsytpecjg gtg xpbdgkynw kqxlqqxsvuar.

Xxinpv ucicanv 2: Uluzw $\ln(8x)-\ln(x+4)=2$ bnezmqd.

Xlszryq sxv wqquhpvjqz:

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

Zrdic yel xctjwxa tquwmjahhov perqujzo:

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

Xpsnaerrvwi xqabj $8x=e^2x+4e^2$ , pv

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

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