Logarithms

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Definition

concept

The logarithm of a positive real number aa in base bb is the exponent cc to which the base bb must be raised to obtain aa:

log⁡ba=c⟺bc=a\log_b a = c \quad \Longleftrightarrow \quad b^c = a

where:

  • a>0a > 0 is the argument of the logarithm,
  • b>0b > 0, with b≠1b \neq 1, is the base,
  • c∈Rc \in \mathbb{R} is the logarithm or exponent.

Logarithms in the real numbers

  • Domain: a∈(0,+∞)a \in (0, +\infty)
  • Range: c∈Rc \in \mathbb{R}
  • Restrictions:

    b>0,b≠1 b > 0, \quad b \neq 1

f(x)=log⁡2xf(x)=\log_2 x

Graph of the base-2 logarithm function. Note that the function is strictly increasing and is only defined for positive numbers.

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Remember

The logarithm in the real numbers is only defined for positive arguments.
For example,

log⁡2(−4)\log_2(-4)

does not exist because there is no real number xx such that

2x=−4.2^x = -4.

Every real power of a positive base is always positive.
Therefore,

log⁡2(−4) is not defined in R.\log_2(-4)\ \text{is not defined in } \mathbb{R}.

General properties of logarithms

The logarithm of the base equals one

log⁡bb=1\boxed{\log_b b = 1}

  • log⁡22=1\log_2 2 = 1
  • log⁡1010=1\log_{10} 10 = 1
  • log⁡55=1\log_5 5 = 1
  • log⁡ee=1\log_e e = 1
  • log⁡100100=1\log_{100} 100 = 1
  • log⁡33=1\log_{\sqrt{3}} \sqrt{3} = 1

Logarithm of 1 in any base is zero

log⁡b1=0\boxed{\log_b 1 = 0}

  • log⁡21=0\log_2 1 = 0
  • log⁡101=0\log_{10} 1 = 0
  • log⁡51=0\log_5 1 = 0
  • log⁡e1=0(i.e., ln⁡1=0)\log_e 1 = 0 \quad \text{(i.e., } \ln 1 = 0)
  • log⁡1001=0\log_{100} 1 = 0
  • log⁡31=0\log_{\sqrt{3}} 1 = 0

Logarithm of a product in the same base

log⁡b(xy)=log⁡bx+log⁡by\boxed{\log_b (xy) = \log_b x + \log_b y}

  • log⁡2(4⋅8)=log⁡24+log⁡28\log_2 (4 \cdot 8) = \log_2 4 + \log_2 8
  • log⁡3(9⋅27)=log⁡39+log⁡327\log_3 (9 \cdot 27) = \log_3 9 + \log_3 27
  • log⁡10(5⋅2)=log⁡105+log⁡102\log_{10} (5 \cdot 2) = \log_{10} 5 + \log_{10} 2
  • ln⁡(e⋅e2)=ln⁡e+ln⁡e2\ln (e \cdot e^2) = \ln e + \ln e^2
  • log⁡5(25⋅125)=log⁡525+log⁡5125\log_5 (25 \cdot 125) = \log_5 25 + \log_5 125
  • log⁡6(6⋅36)=log⁡66+log⁡636\log_6 (6 \cdot 36) = \log_6 6 + \log_6 36

Caution

log⁡b(x+y)≠log⁡bx+log⁡by\log_b (x+y) \not = \log_b x + \log_b y

Logarithm of a quotient in the same base

log⁡b(xy)=log⁡bx−log⁡by\boxed{\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y}

  • log⁡2(82)=log⁡28−log⁡22\log_2 \left(\frac{8}{2}\right) = \log_2 8 - \log_2 2
  • log⁡3(279)=log⁡327−log⁡39\log_3 \left(\frac{27}{9}\right) = \log_3 27 - \log_3 9
  • log⁡10(100010)=log⁡101000−log⁡1010\log_{10} \left(\frac{1000}{10}\right) = \log_{10} 1000 - \log_{10} 10
  • ln⁡(e5e2)=ln⁡e5−ln⁡e2\ln \left(\frac{e^5}{e^2}\right) = \ln e^5 - \ln e^2
  • log⁡5(12525)=log⁡5125−log⁡525\log_5 \left(\frac{125}{25}\right) = \log_5 125 - \log_5 25
  • log⁡6(366)=log⁡636−log⁡66\log_6 \left(\frac{36}{6}\right) = \log_6 36 - \log_6 6

Logarithm of a power

log⁡b(xn)=nlog⁡bx\boxed{\log_b (x^n) = n \log_b x}

  • log⁡2(43)=3log⁡24\log_2 (4^3) = 3 \log_2 4
  • log⁡3(92)=2log⁡39\log_3 (9^2) = 2 \log_3 9
  • log⁡10(1004)=4log⁡10100\log_{10} (100^4) = 4 \log_{10} 100
  • ln⁡(e7)=7ln⁡e\ln (e^7) = 7 \ln e
  • log⁡5(253)=3log⁡525\log_5 (25^3) = 3 \log_5 25
  • log⁡6(65)=5log⁡66\log_6 (6^5) = 5 \log_6 6

Observation

log⁡bnx=(log⁡bx)n≠log⁡bxn\log^n_b x=(\log_b x)^n \not = \log_b x^n

Logarithm of a root

log⁡bxn=1nlog⁡bx\boxed{\log_b \sqrt[n]{x} = \frac{1}{n} \log_b x}

  • log⁡283=13log⁡28\log_2 \sqrt[3]{8} = \frac{1}{3} \log_2 8
  • log⁡39=12log⁡39\log_3 \sqrt{9} = \frac{1}{2} \log_3 9
  • log⁡10100004=14log⁡1010000\log_{10} \sqrt[4]{10000} = \frac{1}{4} \log_{10} 10000
  • ln⁡e105=15ln⁡e10\ln \sqrt[5]{e^{10}} = \frac{1}{5} \ln e^{10}
  • log⁡51253=13log⁡5125\log_5 \sqrt[3]{125} = \frac{1}{3} \log_5 125
  • log⁡636=12log⁡636\log_6 \sqrt{36} = \frac{1}{2} \log_6 36

Logarithm with exponential base and argument

log⁡bmxn=nmlog⁡bx\boxed{\log_{b^m} x^n = \frac{n}{m} \log_b x}

  • log⁡2382=23log⁡28\log_{2^3} 8^2 = \frac{2}{3} \log_2 8
  • log⁡1021004=42log⁡10100\log_{10^2} 100^4 = \frac{4}{2} \log_{10} 100
  • log⁡3492=24log⁡39\log_{3^4} 9^2 = \frac{2}{4} \log_3 9
  • log⁡e5e7=75ln⁡e\log_{e^5} e^7 = \frac{7}{5} \ln e
  • log⁡52253=32log⁡525\log_{5^2} 25^3 = \frac{3}{2} \log_5 25
  • log⁡6663=36log⁡66\log_{6^6} 6^3 = \frac{3}{6} \log_6 6

Equivalence of logarithmic expressions

log⁡bx=log⁡bnxn=log⁡bmxm\boxed{\log_{b} x = \log_{b^n} x^n = \log_{\sqrt[m]{b}} \sqrt[m]{x}}

Chain rule

log⁡by⋅log⁡ya⋅log⁡ax=log⁡bx\boxed{\log_{b} y \cdot \log_{y} a \cdot \log_{a} x = \log_{b} x}

  • log⁡24⋅log⁡48⋅log⁡816=log⁡216\log_{2} 4 \cdot \log_{4} 8 \cdot \log_{8} 16 = \log_{2} 16
  • log⁡39⋅log⁡927⋅log⁡273=log⁡33\log_{3} 9 \cdot \log_{9} 27 \cdot \log_{27} 3 = \log_{3} 3
  • log⁡10100⋅log⁡1001000⋅log⁡100010=log⁡1010\log_{10} 100 \cdot \log_{100} 1000 \cdot \log_{1000} 10 = \log_{10} 10
  • log⁡525⋅log⁡25125⋅log⁡125625=log⁡5625\log_{5} 25 \cdot \log_{25} 125 \cdot \log_{125} 625 = \log_{5} 625
  • ln⁡2⋅log⁡2e⋅log⁡e4=ln⁡4\ln 2 \cdot \log_{2} e \cdot \log_{e} 4 = \ln 4
  • log⁡636⋅log⁡366⋅log⁡6216=log⁡6216\log_{6} 36 \cdot \log_{36} 6 \cdot \log_{6} 216 = \log_{6} 216

Unit product

log⁡bx⋅log⁡xb=1\boxed{\log_{b} x \cdot \log_{x} b = 1}

log⁡bx=1log⁡xb\log_{b} x = \frac{1}{\log_{x} b}

Change of base

log⁡ba=log⁡kalog⁡kb \boxed{\log_b a = \frac{\log_k a}{\log_k b}}

(k>0,k≠1)(k > 0, k \neq 1)

Swap rule

xlog⁡by=ylog⁡bx\boxed{x^{\log_b y} = y^{\log_b x}}

Special properties

blog⁡bx=x\boxed{b^{\log_b x} = x}

Cologarithm

Defined as the logarithm of the reciprocal of a number:

colog⁡bx=log⁡b(1x)=−log⁡bx\boxed{\operatorname{colog}_b x = \log_b \left(\frac{1}{x}\right) = -\log_b x}

x>0,b>0∧b≠1x > 0, b > 0 \land b \neq 1

  • colog⁡102=log⁡10(12)=−log⁡102≈−0.3010\operatorname{colog}_{10} 2 = \log_{10} \left(\frac{1}{2}\right) = -\log_{10} 2 \approx -0.3010
  • colog⁡28=log⁡2(18)=−log⁡28=−3\operatorname{colog}_{2} 8 = \log_{2} \left(\frac{1}{8}\right) = -\log_{2} 8 = -3
  • colog⁡e5=ln⁡(15)=−ln⁡5≈−1.6094\operatorname{colog}_{e} 5 = \ln \left(\frac{1}{5}\right) = -\ln 5 \approx -1.6094
  • colog⁡39=log⁡3(19)=−log⁡39=−2\operatorname{colog}_{3} 9 = \log_{3} \left(\frac{1}{9}\right) = -\log_{3} 9 = -2
  • colog⁡525=log⁡5(125)=−log⁡525=−2\operatorname{colog}_{5} 25 = \log_{5} \left(\frac{1}{25}\right) = -\log_{5} 25 = -2
  • colog⁡416=log⁡4(116)=−log⁡416=−2\operatorname{colog}_{4} 16 = \log_{4} \left(\frac{1}{16}\right) = -\log_{4} 16 = -2

Antilogarithm

It is the inverse operation of the logarithm:

antilog⁡bx=bx\boxed{\operatorname{antilog}_b x = b^x}

b>0,b≠1∧x∈Rb > 0, b \neq 1 \land x \in \mathbb{R}

  • antilog⁡102=102=100\operatorname{antilog}_{10} 2 = 10^2 = 100
  • antilog⁡23=23=8\operatorname{antilog}_{2} 3 = 2^3 = 8
  • antilog⁡e1=e1≈2.7183\operatorname{antilog}_{e} 1 = e^1 \approx 2.7183
  • antilog⁡34=34=81\operatorname{antilog}_{3} 4 = 3^4 = 81
  • antilog⁡50=50=1\operatorname{antilog}_{5} 0 = 5^0 = 1
  • antilog⁡10(−1)=10−1=0.1\operatorname{antilog}_{10} (-1) = 10^{-1} = 0.1

antilog⁡b(log⁡bx)=x\operatorname{antilog}_b (\log_{b}x) = x

log⁡b(antilog⁡bx)=x\log_{b} (\operatorname{antilog}_b x) = x

Logarithmic systems

  1. Common logarithm (base 10):

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

  2. Natural logarithm (base ee):

    log⁡ex=ln⁡x;x>0 \log_e x = \ln x ; \quad x > 0

    ln⁡e=1 \ln e = 1

System conversion

log⁡ba=ln⁡aln⁡b=log⁡alog⁡b\log_b a = \frac{\ln a}{\ln b} = \frac{\log a}{\log b}

To convert between bases mm and nn:

log⁡ba=log⁡malog⁡mb=log⁡nalog⁡nb\log_b a = \frac{\log_{m} a}{\log_{m} b} = \frac{\log_{n} a}{\log_{n} b}

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Important

• Common logarithm

log⁡x=log⁡10x\log x = \log_{10} x

• Natural logarithm

ln⁡x=log⁡ex\ln x = \log_e x

where ee is Euler’s number.

Also

ln⁡1=0\ln 1 = 0

ln⁡e=1\ln e = 1

ln⁡(ex)=x\ln(e^x) = x

Logarithmic equations

  1. Basic equation:

    log⁡bf(x)=c⇒f(x)=bc \log_b f(x) = c \quad \Rightarrow \quad f(x) = b^c

  2. Equation with the same base:

    log⁡bf(x)=log⁡bg(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):

    log⁡bf(x)>log⁡bg(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):

    log⁡bf(x)>log⁡bg(x)⇒0<f(x)<g(x) \log_b f(x) > \log_b g(x) \quad \Rightarrow \quad 0 < f(x) < g(x)