Logarithms

This material covers the topic of logarithms within the context of pre-university algebra, including their definition, fundamental properties, systems of logarithms (decimal and natural), change of base, and applications in equations and inequalities. It contains essential formulas, brief proofs, and theoretical examples to strengthen the understanding of logarithmic functions and their use in advanced mathematical problems. Ideal for students preparing for university entrance exams or reinforcing key algebra concepts.

Definition

Hint

The logarithm of a number $a > 0$ with base $b$ (where $b > 0$ , $b \neq 1$ ) is the exponent $c$ to which $b$ must be raised to obtain $a$ :

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

Logarithms in the Real Numbers

Domain: $a \in (0, +\infty)$
Range: $c \in \mathbb{R}$
Restrictions: $b > 0, \quad b \neq 1$

General Properties of Logarithms

Logarithm of the Base Equals One

\log_b b = 1

Examples

$\log_2 2 = 1$
$\log_{10} 10 = 1$
$\log_5 5 = 1$
$\log_e e = 1$
$\log_{100} 100 = 1$
$\log_{\sqrt{3}} \sqrt{3} = 1$

Logarithm of 1 in Any Base Is Zero

\log_b 1 = 0

Examples

$\log_2 1 = 0$
$\log_{10} 1 = 0$
$\log_5 1 = 0$
$\log_e 1 = 0$ (i.e., $\ln 1 = 0$ )
$\log_{100} 1 = 0$
$\log_{\sqrt{3}} 1 = 0$

Logarithm of a Product (Same Base)

\log_b (xy) = \log_b x + \log_b y

Examples

$\log_2 (4 \cdot 8) = \log_2 4 + \log_2 8$
$\log_3 (9 \cdot 27) = \log_3 9 + \log_3 27$
$\log_{10} (5 \cdot 2) = \log_{10} 5 + \log_{10} 2$
$\ln (e \cdot e^2) = \ln e + \ln e^2$
$\log_5 (25 \cdot 125) = \log_5 25 + \log_5 125$
$\log_6 (6 \cdot 36) = \log_6 6 + \log_6 36$

Logarithm of a Quotient (Same Base)

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

Examples

$\log_2 \left(\frac{8}{2}\right) = \log_2 8 - \log_2 2$
$\log_3 \left(\frac{27}{9}\right) = \log_3 27 - \log_3 9$
$\log_{10} \left(\frac{1000}{10}\right) = \log_{10} 1000 - \log_{10} 10$
$\ln \left(\frac{e^5}{e^2}\right) = \ln e^5 - \ln e^2$
$\log_5 \left(\frac{125}{25}\right) = \log_5 125 - \log_5 25$
$\log_6 \left(\frac{36}{6}\right) = \log_6 36 - \log_6 6$

Logarithm of a Power

\log_b (x^n) = n \log_b x

Examples

$\log_2 (4^3) = 3 \log_2 4$
$\log_3 (9^2) = 2 \log_3 9$
$\log_{10} (100^4) = 4 \log_{10} 100$
$\ln (e^7) = 7 \ln e$
$\log_5 (25^3) = 3 \log_5 25$
$\log_6 (6^5) = 5 \log_6 6$

Logarithm of a Root

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

Examples

$\log_2 \sqrt[3]{8} = \frac{1}{3} \log_2 8$
$\log_3 \sqrt{9} = \frac{1}{2} \log_3 9$
$\log_{10} \sqrt[4]{10000} = \frac{1}{4} \log_{10} 10000$
$\ln \sqrt[5]{e^{10}} = \frac{1}{5} \ln e^{10}$
$\log_5 \sqrt[3]{125} = \frac{1}{3} \log_5 125$
$\log_6 \sqrt{36} = \frac{1}{2} \log_6 36$

Logarithm with Exponential Base and Argument

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

Examples

$\log_{2^3} 8^2 = \frac{2}{3} \log_2 8$
$\log_{10^2} 100^4 = \frac{4}{2} \log_{10} 100$
$\log_{3^4} 9^2 = \frac{2}{4} \log_3 9$
$\log_{e^5} e^7 = \frac{7}{5} \ln e$
$\log_{5^2} 25^3 = \frac{3}{2} \log_5 25$
$\log_{6^6} 6^3 = \frac{3}{6} \log_6 6$

Equivalence of Logarithmic Expressions

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

Chain Rule

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

Examples

$\log_{2} 4 \cdot \log_{4} 8 \cdot \log_{8} 16 = \log_{2} 16$
$\log_{3} 9 \cdot \log_{9} 27 \cdot \log_{27} 3 = \log_{3} 3$
$\log_{10} 100 \cdot \log_{100} 1000 \cdot \log_{1000} 10 = \log_{10} 10$
$\log_{5} 25 \cdot \log_{25} 125 \cdot \log_{125} 625 = \log_{5} 625$
$\ln 2 \cdot \log_{2} e \cdot \log_{e} 4 = \ln 4$
$\log_{6} 36 \cdot \log_{36} 6 \cdot \log_{6} 216 = \log_{6} 216$

Reciprocal Identity

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

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

Change of Base Formula

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

Logarithmic Exchange Rule

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

Special Property

b^{\log_b x} = x

Cologarithm

Defined as the logarithm of the reciprocal of a number:

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

Examples

$\operatorname{colog}_{10} 2 = \log_{10} \left(\frac{1}{2}\right) = -\log_{10} 2 \approx -0.3010$
$\operatorname{colog}_{2} 8 = \log_{2} \left(\frac{1}{8}\right) = -\log_{2} 8 = -3$
$\operatorname{colog}_{e} 5 = \ln \left(\frac{1}{5}\right) = -\ln 5 \approx -1.6094$
$\operatorname{colog}_{3} 9 = \log_{3} \left(\frac{1}{9}\right) = -\log_{3} 9 = -2$
$\operatorname{colog}_{5} 25 = \log_{5} \left(\frac{1}{25}\right) = -\log_{5} 25 = -2$
$\operatorname{colog}_{4} 16 = \log_{4} \left(\frac{1}{16}\right) = -\log_{4} 16 = -2$

Antilogarithm

The inverse operation of the logarithm:

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

Examples

$\operatorname{antilog}_{10} 2 = 10^2 = 100$
$\operatorname{antilog}_{2} 3 = 2^3 = 8$
$\operatorname{antilog}_{e} 1 = e^1 \approx 2.7183$
$\operatorname{antilog}_{3} 4 = 3^4 = 81$
$\operatorname{antilog}_{5} 0 = 5^0 = 1$
$\operatorname{antilog}_{10} (-1) = 10^{-1} = 0.1$

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

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

Logarithmic Systems

Common logarithm (base 10):
$\log_{10} x \equiv \log x; \quad x > 0$
Natural logarithm (base $e$ ):
$\log_e x = \ln x; \quad x > 0$ $\ln e = 1$

Base Conversion

\log_b a = \frac{\ln a}{\ln b} = \frac{\log a}{\log b}

To convert between any bases $m$ and $n$ :

\log_b a = \frac{\log_{m} a}{\log_{m} b} = \frac{\log_{n} a}{\log_{n} b}

Logarithmic Equations

Basic equation:
$\log_b f(x) = c \quad \Rightarrow \quad f(x) = b^c$
Same-base equation:
$\log_b f(x) = \log_b g(x) \quad \Rightarrow \quad f(x) = g(x)$

Logarithmic Inequalities

Consider the base:

If $b > 1$ (logarithmic function is increasing):
$\log_b f(x) > \log_b g(x) \quad \Rightarrow \quad f(x) > g(x) > 0$
If $0 < b < 1$ (logarithmic function is decreasing):
$\log_b f(x) > \log_b g(x) \quad \Rightarrow \quad 0 < f(x) < g(x)$