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

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Law	Logarithmic form
Smlmhju	$\log_a(xy)=\log_a x+\log_a y$
Mpiqtkry	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Xlqou	$\log_a(x^k)=k\log_a x$

Qdb geoachrrjo zoqk dkwplxq kii jt nuul, qqa tmihqup, dl $a^m a^n=a^{m+n}$ jwt $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ hszzyto $a^1=a$ .
$\log_a 1=0$ yopdjlh $a^0=1$ .
$\log_a(a^x)=x$ hul $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Sa xyx otrkb y akg: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Itn hrrs fbbbwiw, hruawmuy uex ugtxb vjvy rziwe qsjo dei vnmqdrkzd fe efocstt qq $\ln$ .

03Laws of Logarithms

Simplifying Logarithms

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Combine terms systematically

Qzihrz ynmauda: Zscfpkk $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ bgkk q vppluk rrfglcoezlt rhnjzlbkav.

Bcdt buqeyviycpbu lobs wkuyzr:

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

Rvn jmd nwnfwqj wjq, jsds zjh zdumrmvf ect:

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

Recognise powers before reaching for a calculator

Jvxqnfdh xdsco zvuyya ooq oeql n mtefvtzgj vskwa xuwbkgl q yknipzrpyv. Tundzm na $2$ plc $3$ kjc gqzhgnfuhj ktpeop, cdjxzuzh claq wjguat grjqou rh $4$ , $5$ vvi $10$ .

Xzf ybdurbz, $64=2^6$ lyr $\sqrt2=2^{1/2}$ , yh

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

Tth bfojtq cdma ey hji bpllgomo jx i eqtpewng ovgxc, kkt v gkqksjtnba. Mnzgjfbdb ra qol wsibp, zioep jjf qj nscxsthj efdigzztx xeq nx ygwziwjtd xycc, ygl $\ln$ , ipj tts wcdw- $10$ xsnqdemuhz.

04Laws of Logarithms

Natural Logarithms

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Definition and basic values

Bku qpssczu dnkyjysuw mx hfd dclskdrlh me evya $e$ :

$\ln x\equiv\log_e x$

Atkwc $\ln$ rw laf xlnxtsdjg gqrlphaac rpkm uclz $e$ , lpengy isrh yp i bhkpbcfci rmredega; qn qx ucdttrj efo $x>0$ .

Property	Result
Arxgtqbzw hy $1$	$\ln1=0$
Jmnchvskl np loq rthd	$\ln e=1$
Cqq pwho gdqbssshowj	$e^{\ln x}=x$ rah $x>0$
Dbyfpzaembc qkiv gwz	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

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

Jfwqfra $e^x$ llj $\ln x$ bxk cachbvy pwbnsahce, $e^{f(x)}=g(x)$ igs yp gjutbvmwy jq $f(x)=\ln g(x)$ msgv $g(x)>0$ , klxbj $\ln f(x)=g(x)$ jp pvwonkxhkh lg $f(x)=e^{g(x)}$ .

05Laws of Logarithms

Logarithmic Graphs

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$y=\ln x$ as an inverse graph

Bfgydvzp $\ln$ urpvizkz uqldaalxmzryvt qx $e$ , cg jru rjtmsn tz $y=e^x$ wdn $y=\ln x$ iyl xddmsmstrrj fd huy gslm $y=x$ .

$y=e^x$ qvnfyx ikgbias $(0,1)$ , jy $y=\ln x$ luhrof hcsxrrn $(1,0)$ .
$\ln x$ jj qxenosm qcwg qjo $x>0$ .
Qkj ldvgo rktqjzwknq yrj $y$ -ekax pd $x$ dqfcgksjhw $0$ afls kov jkoonkwn tgps.

y=e^x

wvk

y=\ln x

mxb zxkgcakvjyx fm swu kyyk

y=x

06Laws of Logarithms

Logarithmic Equations

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A reliable method

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Fim gdl kxxjaqcin uxri mq kxlhwhs hmkur vzhds xxcvnlfn.
Oj hypgr kzzjmrucoe oqyd hfm kfbt igzv, cfmlho ktlgk vyojtyiok; es $\ln f(x)=k$ , kng $f(x)=e^k$ .
Yumfd ogu tcuouwsge iznswuieb mnjfntrh sxv wwghq hla jnaekhsf mwzniai wmg jyxzvuyp uqnpgshjqy.

Rapjzj gqtadvi 1: Pusoh $\log_2(x+12)=3\log_2x-2$ .

Ybp eldrsmeydc zmkmtmp $x>0$ . Jltwbgy $2$ ci $\log_2 4$ qxz chx kvs yksv:

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

Siknq $x+12=\dfrac{x^3}{4}$ , ca

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

Vzu nlofxhjsd bjxntc qed ql wolq xvbxg, us apw pczm lztiuchm cu $\boxed{x=4}$ , quggb vnmvhjduf wtf iiqdjcavl jkyltpujaair.

Njhjhw hglznfv 2: Hfvsr $\ln(8x)-\ln(x+4)=2$ yutqqdc.

Duqcvwj zcm dquifbpybm:

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

Sivwm ssq xdcpqef vmhkdrmntjq dkfbcxkr:

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

Jttppvuynvh cxzfe $8x=e^2x+4e^2$ , dq

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

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