Basic Mathematics - Lecture 02 (JEE 2027)

Flashcards covering key definitions, special modulus properties, logarithm properties, logarithmic inequalities, and exponential functions from Basic Mathematics Lecture 02 by Sachin Jakhar Sir.

Domain of Logarithm

A logarithmic expression $\log_B(A)$ is defined if and only if $A > 0$, $B > 0$, and $B \neq 1$.

Modulus Inequality for $|x| < a$

Assuming $a > 0$, the inequality $|x| < a$ simplifies to $-a < x < a$.

Modulus Inequality for $|x| > a$

Assuming $a > 0$, the inequality $|x| > a$ simplifies to $x < -a$ union $x > a$.

Special Modulus Property 1

The equation $|\alpha| + |\beta| = |\alpha + \beta|$ holds true if and only if $\alpha \beta \ge 0$.

Special Modulus Property 2

The inequality $|\alpha| + |\beta| > |\alpha + \beta|$ holds true if and only if $\alpha \beta < 0$.

Special Modulus Property 3

The equation $|\alpha| + |\beta| = |\alpha - \beta|$ holds true if and only if $\alpha \beta \le 0$.

Logarithm Product Property

$$\log_a(m) + \log_a(n) = \log_a(m n)$$

Logarithm Quotient Property

$$\log_a(m) - \log_a(n) = \log_a\left(\frac{m}{n}\right)$$

Logarithm Power Property

$$\log_{a^q}(m^p) = \frac{p}{q} \log_a(m)$$

Base Change Property of Logarithms

$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, and reciprocally, $\log_b(a) = \frac{1}{\log_a(b)}$.

Logarithm Power Exchange Property

$$a^{\log_c(b)} = b^{\log_c(a)}$$

Natural Logarithm

A logarithm with base $e$ (where $e \approx 2.72$), denoted as $\ln(x) = \log_e(x)$.

Logarithmic Inequality Rule (Base $> 1$)

When applying or removing a logarithm with a base greater than $1$, the direction of the inequality remains unchanged.

Logarithmic Inequality Rule ($0 < \text{Base} < 1$)

When applying or removing a logarithm with a base between $0$ and $1$, the direction of the inequality gets reversed.

Square Root of a Squared Variable

For any real variable $x$, $\sqrt{x^2} = |x|$.

Sign of Exponential Function

For an exponential function $f(x) = a^x$ with $a > 0$ and $a \neq 1$, the output is strictly positive ($a^x > 0$ for all $x \in \mathbb{R}$).

Exponential Inequality Rule (Base $> 1$)

If $a^x \ge a^y$ and $a > 1$, then $x \ge y$.

Exponential Inequality Rule ($0 < \text{Base} < 1$)

If $a^x \ge a^y$ and $0 < a < 1$, then $x \le y$.