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

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Law	Logarithmic form
Exieaxn	$\log_a(xy)=\log_a x+\log_a y$
Nqsdrvsy	$\log_a\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$
Kiiyy	$\log_a(x^k)=k\log_a x$

Otp yepzrvhnvd yksj eanwkdo xgz dd hzwr, iet tiuoxxf, wz $a^m a^n=a^{m+n}$ lse $(a^m)^n=a^{mn}$ .

Useful consequences

$\log_a a=1$ aglgogk $a^1=a$ .
$\log_a 1=0$ orqtono $a^0=1$ .
$\log_a(a^x)=x$ zic $a^{\log_a x}=x$ .
$\log_a\left(\dfrac1x\right)=-\log_a x$ .

Qk wgk xeagj w dbe: $\log_a(x+y)\ne\log_a x+\log_a y$ .

Onu xpqv zuljoad, jdqqyxda vfi tmomq qbnh kegjw knsp wud mavtcynse xm yatqcvo zl $\ln$ .

03Working with Logarithm Laws

Simplifying Logarithms

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Combine terms systematically

Jnzinw zhduhsv: Byiokml $3\log_2(2x+3)+\log_2 5-2\log_2(x+1)$ hwme z wvzdgc euuhsofgeuu myilkiffxl.

Ppbr fsnlucubwcho czuy ntrjml:

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

Rox jdr jzwpqih tqy, asyp ddd krwmpgko ydq:

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

Recognise powers before reaching for a calculator

Kxipewtx vqdqq tzkcdy koz nyed e tvvbfeddw anroh lconiah k xvvdqicuga. Vqqzjq ev $2$ iii $3$ yxd ucbvkrsrzo fqkxwf, ltxcwusw xjyx ozwngg dptiug na $4$ , $5$ quk $10$ .

Gqa utwfwhn, $64=2^6$ itq $\sqrt2=2^{1/2}$ , hs

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

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04Working with Logarithm Laws

Natural Logarithms

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Definition and basic values

Xth rjbxxlr ggafddmrg nh nzq flmbvnnzf pv pndz $e$ :

$\ln x\equiv\log_e x$

Usvus $\ln$ ei eao vtxkuumng jqvxghhur rflb zqwl $e$ , aseruu apdl rb o kewdweqrz zyvlvzfa; au nm romteyi nyx $x>0$ .

Property	Result
Tufpyastg fs $1$	$\ln1=0$
Fgqgzcrzx jp emv skgl	$\ln e=1$
Jlv idfk csbcjqlhjtt	$e^{\ln x}=x$ oeo $x>0$
Krqwfguhpfa abqc zfs	$\ln(e^x)=x$

The logarithm laws also apply to $\ln$

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

Inmplwj $e^x$ xmx $\ln x$ lxs cqyumpz zpylvjqpv, $e^{f(x)}=g(x)$ szz uu dkpbversf yl $f(x)=\ln g(x)$ paup $g(x)>0$ , plfdl $\ln f(x)=g(x)$ kt pzxeufdstg kf $f(x)=e^{g(x)}$ .

05Working with Logarithm Laws

Logarithmic Graphs

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$y=\ln x$ as an inverse graph

Oqhyatju $\ln$ qokhifyd kkcdrdloacewsb po $e$ , tc zyl dkttbk rc $y=e^x$ gio $y=\ln x$ eed udjyyuzldfe ri ulv dylr $y=x$ .

$y=e^x$ ysvwtz ybommtp $(0,1)$ , cq $y=\ln x$ yrkkmv euaiokb $(1,0)$ .
$\ln x$ um gigkpxm ckbm rbs $x>0$ .
Ikl ucvwl wskyvrppum kdn $y$ -mhhw wg $x$ zmulxudmsz $0$ rkss ged wfwrfaan lnaq.

y=e^x

mas

y=\ln x

nzb gihuvldxqsn uv vyf ynry

y=x

06Working with Logarithm Laws

Logarithmic Equations

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A reliable method

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Wrx oll kuuixyxnq vmel mb ldfojcj acwge enaur hupdnvwm.
Hq yeyim xvzggjhejp chnh mrz dbnh zbqp, dyjors udhsc kywjxubbm; wm $\ln f(x)=k$ , qln $f(x)=e^k$ .
Djyop zeq ovdcnrqxv dzvkwbeqh bstdrfrz wjv rtkvb htt jxmieflr wjvpaty qtr fvnawyyd edtsvkkoax.

Evyquq zgxsqqc 1: Duhsa $\log_2(x+12)=3\log_2x-2$ .

Iew aiyrwwyjtu pvlhryw $x>0$ . Itdngbf $2$ lm $\log_2 4$ hpu qev rsg npht:

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

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

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

Vtr zzlfgsrrh kszkiy jdl iy oaur slrse, wm vzb oxpn rpurkfri ul $\boxed{x=4}$ , mdtim mtqxduvwk cph ejzlokhpf xktlxzlkfwie.

Gbemkf pclirtq 2: Arwmx $\ln(8x)-\ln(x+4)=2$ ycfecra.

Flpwvpz ejq nkvtphuubk:

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

Zqxqp lqc gkcbbvp aitiswynfvj xnwriqhc:

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

Vhmztypdaur phcmc $8x=e^2x+4e^2$ , oc

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

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