Logarithms

Master logarithms with this comprehensive guide! Learn the definition, properties (product, quotient, power rules), change of base, and solving logarithmic equations. Includes examples, cologarithm, antilogarithm, and inequalities. Perfect for students and math enthusiasts!

Definition

Tip

The logarithm of a number a>0a > 0 in base bb (b>0b > 0, b1b \neq 1) is the exponent cc to which bb must be raised to obtain aa:

logba=cbc=a\log_b a = c \quad \Leftrightarrow \quad b^c = a

Logarithms in the Real Numbers

  • Domain: a(0,+)a \in (0, +\infty)
  • Range: cRc \in \mathbb{R}
  • Restrictions:

    b>0,b1b > 0, \quad b \neq 1

General Properties of Logarithms

Important

  1. The logarithm of the base equals one:

    logbb=1\log_b b = 1

Examples

  • log22=1\log_2 2 = 1
  • log1010=1\log_{10} 10 = 1
  • log55=1\log_5 5 = 1
  • logee=1\log_e e = 1
  • log100100=1\log_{100} 100 = 1
  • log33=1\log_{\sqrt{3}} \sqrt{3} = 1
  1. The logarithm of 1 in any base is zero:

    logb1=0\log_b 1 = 0

Examples

  • log21=0\log_2 1 = 0
  • log101=0\log_{10} 1 = 0
  • log51=0\log_5 1 = 0
  • loge1=0\log_e 1 = 0 (i.e., ln1=0\ln 1 = 0)
  • log1001=0\log_{100} 1 = 0
  • log31=0\log_{\sqrt{3}} 1 = 0
  1. Logarithm of a product in the same base:

    logb(xy)=logbx+logby\log_b (xy) = \log_b x + \log_b y

Examples

  • log2(48)=log24+log28\log_2 (4 \cdot 8) = \log_2 4 + \log_2 8
  • log3(927)=log39+log327\log_3 (9 \cdot 27) = \log_3 9 + \log_3 27
  • log10(52)=log105+log102\log_{10} (5 \cdot 2) = \log_{10} 5 + \log_{10} 2
  • ln(ee2)=lne+lne2\ln (e \cdot e^2) = \ln e + \ln e^2
  • log5(25125)=log525+log5125\log_5 (25 \cdot 125) = \log_5 25 + \log_5 125
  • log6(636)=log66+log636\log_6 (6 \cdot 36) = \log_6 6 + \log_6 36
  1. Logarithm of a quotient in the same base:

    logb(xy)=logbxlogby\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y

Examples

  • log2(82)=log28log22\log_2 \left(\frac{8}{2}\right) = \log_2 8 - \log_2 2
  • log3(279)=log327log39\log_3 \left(\frac{27}{9}\right) = \log_3 27 - \log_3 9
  • log10(100010)=log101000log1010\log_{10} \left(\frac{1000}{10}\right) = \log_{10} 1000 - \log_{10} 10
  • ln(e5e2)=lne5lne2\ln \left(\frac{e^5}{e^2}\right) = \ln e^5 - \ln e^2
  • log5(12525)=log5125log525\log_5 \left(\frac{125}{25}\right) = \log_5 125 - \log_5 25
  • log6(366)=log636log66\log_6 \left(\frac{36}{6}\right) = \log_6 36 - \log_6 6
  1. Logarithm of a power:

    logb(xn)=nlogbx\log_b (x^n) = n \log_b x

Examples

  • log2(43)=3log24\log_2 (4^3) = 3 \log_2 4
  • log3(92)=2log39\log_3 (9^2) = 2 \log_3 9
  • log10(1004)=4log10100\log_{10} (100^4) = 4 \log_{10} 100
  • ln(e7)=7lne\ln (e^7) = 7 \ln e
  • log5(253)=3log525\log_5 (25^3) = 3 \log_5 25
  • log6(65)=5log66\log_6 (6^5) = 5 \log_6 6
  1. Logarithm of a root:

    logbxn=1nlogbx\log_b \sqrt[n]{x} = \frac{1}{n} \log_b x

Examples

  • log283=13log28\log_2 \sqrt[3]{8} = \frac{1}{3} \log_2 8
  • log39=12log39\log_3 \sqrt{9} = \frac{1}{2} \log_3 9
  • log10100004=14log1010000\log_{10} \sqrt[4]{10000} = \frac{1}{4} \log_{10} 10000
  • lne105=15lne10\ln \sqrt[5]{e^{10}} = \frac{1}{5} \ln e^{10}
  • log51253=13log5125\log_5 \sqrt[3]{125} = \frac{1}{3} \log_5 125
  • log636=12log636\log_6 \sqrt{36} = \frac{1}{2} \log_6 36
  1. Logarithm with exponential base and argument:

    logbmxn=nmlogbx\log_{b^m} x^n = \frac{n}{m} \log_b x

Examples

  • log2382=23log28\log_{2^3} 8^2 = \frac{2}{3} \log_2 8
  • log1021004=42log10100\log_{10^2} 100^4 = \frac{4}{2} \log_{10} 100
  • log3492=24log39\log_{3^4} 9^2 = \frac{2}{4} \log_3 9
  • loge5e7=75lne\log_{e^5} e^7 = \frac{7}{5} \ln e
  • log52253=32log525\log_{5^2} 25^3 = \frac{3}{2} \log_5 25
  • log6663=36log66\log_{6^6} 6^3 = \frac{3}{6} \log_6 6
  1. Equivalence of logarithmic expressions:

    logbx=logbnxn=logbmxm\log_{b} x = \log_{b^n} x^n = \log_{\sqrt[m]{b}} \sqrt[m]{x}

  2. Chain rule:

    logbylogyalogax=logbx\log_{b} y \cdot \log_{y} a \cdot \log_{a} x = \log_{b} x

Examples

  • log24log48log816=log216\log_{2} 4 \cdot \log_{4} 8 \cdot \log_{8} 16 = \log_{2} 16
  • log39log927log273=log33\log_{3} 9 \cdot \log_{9} 27 \cdot \log_{27} 3 = \log_{3} 3
  • log10100log1001000log100010=log1010\log_{10} 100 \cdot \log_{100} 1000 \cdot \log_{1000} 10 = \log_{10} 10
  • log525log25125log125625=log5625\log_{5} 25 \cdot \log_{25} 125 \cdot \log_{125} 625 = \log_{5} 625
  • ln2log2eloge4=ln4\ln 2 \cdot \log_{2} e \cdot \log_{e} 4 = \ln 4
  • log636log366log6216=log6216\log_{6} 36 \cdot \log_{36} 6 \cdot \log_{6} 216 = \log_{6} 216
  1. Unit product:

logbxlogxb=1\log_{b} x \cdot \log_{x} b = 1

logbx=1logxb\log_{b} x = \frac{1}{\log_{x} b}

  1. Change of base:

logba=logkalogkb(k>0,k1)\log_b a = \frac{\log_k a}{\log_k b} \quad (k > 0, k \neq 1)

  1. Exchange rule:

xlogby=ylogbxx^{\log_b y} = y^{\log_b x}

  1. Special properties:

blogbx=xb^{\log_b x} = x

Cologarithm

Tip

Defined as the logarithm of the reciprocal of a number:

cologbx=logb(1x)=logbx;x>0,b>0b1\operatorname{colog}_b x = \log_b \left(\frac{1}{x}\right) = -\log_b x; \quad x > 0, b > 0 \land b \neq 1

Examples

  • colog102=log10(12)=log1020.3010\operatorname{colog}_{10} 2 = \log_{10} \left(\frac{1}{2}\right) = -\log_{10} 2 \approx -0.3010
  • colog28=log2(18)=log28=3\operatorname{colog}_{2} 8 = \log_{2} \left(\frac{1}{8}\right) = -\log_{2} 8 = -3
  • cologe5=ln(15)=ln51.6094\operatorname{colog}_{e} 5 = \ln \left(\frac{1}{5}\right) = -\ln 5 \approx -1.6094
  • colog39=log3(19)=log39=2\operatorname{colog}_{3} 9 = \log_{3} \left(\frac{1}{9}\right) = -\log_{3} 9 = -2
  • colog525=log5(125)=log525=2\operatorname{colog}_{5} 25 = \log_{5} \left(\frac{1}{25}\right) = -\log_{5} 25 = -2
  • colog416=log4(116)=log416=2\operatorname{colog}_{4} 16 = \log_{4} \left(\frac{1}{16}\right) = -\log_{4} 16 = -2

Antilogarithm

Tip

The inverse operation of the logarithm:

antilogbx=bx;b>0,b1xR\operatorname{antilog}_b x = b^x; \quad b > 0, b \neq 1 \land x \in \mathbb{R}

Examples

  • antilog102=102=100\operatorname{antilog}_{10} 2 = 10^2 = 100
  • antilog23=23=8\operatorname{antilog}_{2} 3 = 2^3 = 8
  • antiloge1=e12.7183\operatorname{antilog}_{e} 1 = e^1 \approx 2.7183
  • antilog34=34=81\operatorname{antilog}_{3} 4 = 3^4 = 81
  • antilog50=50=1\operatorname{antilog}_{5} 0 = 5^0 = 1
  • antilog10(1)=101=0.1\operatorname{antilog}_{10} (-1) = 10^{-1} = 0.1

antilogb(logbx)=x\operatorname{antilog}_b (\log_{b}x) = x

logb(antilogbx)=x\log_{b} (\operatorname{antilog}_b x) = x

Logarithm Systems

  1. Common logarithm (base 10):

    log10xlogx;x>0\log_{10} x \equiv \log x; \quad x > 0

  2. Natural logarithm (base ee):

    logex=lnx;x>0\log_e x = \ln x; \quad x > 0

    lne=1\ln e = 1

Conversion Between Systems

logba=lnalnb=logalogb\log_b a = \frac{\ln a}{\ln b} = \frac{\log a}{\log b}

To convert between bases mm and nn:

logba=logmalogmb=lognalognb\log_b a = \frac{\log_{m} a}{\log_{m} b} = \frac{\log_{n} a}{\log_{n} b}

Logarithmic Equations

  1. Basic equation:

    logbf(x)=cf(x)=bc\log_b f(x) = c \quad \Rightarrow \quad f(x) = b^c

  2. Same-base equation:

    logbf(x)=logbg(x)f(x)=g(x)\log_b f(x) = \log_b g(x) \quad \Rightarrow \quad f(x) = g(x)

Logarithmic Inequalities

Consider the base:

  1. If b>1b > 1 (increasing function):

    logbf(x)>logbg(x)f(x)>g(x)>0\log_b f(x) > \log_b g(x) \quad \Rightarrow \quad f(x) > g(x) > 0

  2. If 0<b<10 < b < 1 (decreasing function):

    logbf(x)>logbg(x)0<f(x)<g(x)\log_b f(x) > \log_b g(x) \quad \Rightarrow \quad 0 < f(x) < g(x)