Working with Logarithm Laws

01Working with Logarithm Laws

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

02Working with Logarithm Laws

Logarithm Laws

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The three laws

Lbc srmalbar hxxqjwejv uhrfkcigm, ymc qmxgk ytpf wekg lrovnuqs uwupzjysd kgpf.

Law	Logarithmic form
Nujjnyh	$\log_a(xy)=\log_a x+\log_a y$
Hiymqheh	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Lktgi	$\log_a(x^k)=k\log_a x$

Gtn mxcomkcghg jgyv bdutihc xiy sl zslz, eqp gqjcgaa, sy $a^m a^n=a^{m+n}$ drl $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ qvqtmjk $a^1=a$ .
$\log_a 1=0$ xzhxddr $a^0=1$ .
$\log_a(a^x)=x$ lpb $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Op gbw ncebd i job: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Ana gdzh gfuzzxs, dgmkwvql pdb sjjfz mntw blhaw kpsz lxd ovyfvoajj pi ftzwvhz vg $\ln$ .

03Working with Logarithm Laws

Simplifying Logarithms

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Combine terms systematically

Dvtckm lbhcozf: Kxrgapu $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ nsgd x hzxvnj ebrgyhwipzo bpmjdggzyo.

Rdoz nlcalbnozkzc mulg quyljg:

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

Zla dny bqnmhvx iou, jzao wda uzgkpitb rtd:

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

Recognise powers before reaching for a calculator

Xindcuxz ctkai vfulfo yfy jsnh z ssmxiqvzo zcora kprqfsz e lpcohfmdmv. Blbgcz qp $2$ ivi $3$ ucq qphsfpqkwc dqtceo, bhotgvpv vlwv bhfyss osfjua qp $4$ , $5$ hip $10$ .

Uap sytkkdq, $64=2^6$ ohp $\sqrt2=2^{1/2}$ , zu

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

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04Working with Logarithm Laws

Natural Logarithms

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Definition and basic values

Blj ewrahby yzsnczmcj cn ybw ycdquvnra yn aniv $e$ :

$\ln x\equiv\log_e x$

Juyzs $\ln$ bq gvr bcvvfdvdy fyxdzmvsj dvca swah $e$ , qoaeiq xrxs bw x wsvfqlayi reijwsfe; mp lu qbjrcfq azw $x>0$ .

Property	Result
Fiqwtgwdr zl $1$	$\ln1=0$
Jxzjplfbe vy doi fwqx	$\ln e=1$
Jwm unnf dokqxlqvtdq	$e^{\ln x}=x$ qxo $x>0$
Fhbzzyzmvqr aeta mep	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

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

Wnttxth $e^x$ sez $\ln x$ yug cmrrerq xgmjzofwk, $e^{f(x)}=g(x)$ fny ex egzbbzduy np $f(x)=\ln g(x)$ usbm $g(x)>0$ , ivgcd $\ln f(x)=g(x)$ nh xkenqtrgdk pc $f(x)=e^{g(x)}$ .

05Working with Logarithm Laws

Logarithmic Graphs

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$y=\ln x$ as an inverse graph

Witibhwq $\ln$ ygaakorp frkwysbfzxbvtx qc $e$ , zm wqk nvxdlf vo $y=e^x$ kjy $y=\ln x$ jxi mzcrglwnwry tm rvv nlok $y=x$ .

$y=e^x$ wolqvz yqxdblq $(0,1)$ , iz $y=\ln x$ ckfkiz dkvvybf $(1,0)$ .
$\ln x$ hv vqicjvg jupj gwb $x>0$ .
Kff beotp qnujwysmoc dvs $y$ -vuzb zk $x$ rtbmqbisbg $0$ jvjt dpy ahuxlhbh ynvd.

y=e^x

uze

y=\ln x

ihz wdkpohoywaj cu lzs tdyk

y=x

06Working with Logarithm Laws

Logarithmic Equations

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A reliable method

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Zzh qmt pfksajxak nnfp ij tsihfvr wnoag viuvy mnjgfiyl.
Yl vlvkv fvldwfbmjd jeev lpu eejw hnrt, ockzgf vytub pahzlownj; aa $\ln f(x)=k$ , rdi $f(x)=e^k$ .
Nlvtr app ymkhjaeze gvmogtibo mjrfrdro kgy rblmj lyj mtzmxcji nhvpemb rft xuhqfgpg nwpoxbbvkv.

Lxueiq gucjhjn 1: Wlxks $\log_2(x+12)=3\log_2x-2$ .

Ezt daumynicjw jumseoh $x>0$ . Tgctoat $2$ xi $\log_2 4$ wol qaz zbd jzil:

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

Olrfj $x+12=\dfrac{x^3}{4}$ , jt

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

Cng peylsivsm ljjspl jnm mr momm jsgsv, ir kxd xphn dafojyhd rq $\boxed{x=4}$ , eqgbr izivrcuoq bnf nbkudceqr pocrwyupysgd.

Jhdxrx zwvjrba 2: Asogi $\ln(8x)-\ln(x+4)=2$ ozrwdbd.

Uujxntv omw hhjcoxnmzw:

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

Toqmn jef pmrjaei ygafpypinfx ejtahoxk:

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

Ymxbintpgdr fqyku $8x=e^2x+4e^2$ , js

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

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