Logarithm Formula Sheet

Logarithm is a mathematical function that helps to find the exponent (or power) to which a base number must be raised to obtain a given value.

Neetesh Kumar | May 10, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Properties and Formulas

1. Definition:

The logarithm of the number N to the base ‘a’ is the exponent indicating the power to which the base ‘a’ must be raised to obtain the number N. This number is designated as log$_a$N.

2. Properties and Formulas:

$\bold{(a)}$ log$_a$N = x, read as log of N to the base a $\iff$ $a^x = N$.
If a = 10, then we write log N or log$_{10}$N and if a = e, we write ln N or ln$_e$N (Natural log)

$\bold{(b)}$ Necessary conditions : N > 0 ; a > 0 ; a $\ne$ 1

$\bold{(c)}$ log$_a$1 = 0

$\bold{(d)}$ log$_a$a = 1

$\bold{(e)}$ log$_\frac{1}{a}$a = -1

$\bold{(f)}$ log$_a$x.y = log$_a$x + log$_a$y; x, y > 0

$\bold{(g)}$ log$_a(\frac{x}{y})$ = log$_a$x - log$_a$y; x, y > 0

$\bold{(h)}$ log$_ax^p$ = plog$_a$x ; x > 0

$\bold{(i)}$ log$_{a^q}$x = $\frac{1}{q}$log$_a$x ; x > 0

$\bold{(j)}$ log$_a$x = $\frac{1}{log_ax}$ ; x > 0, x $\ne$ 1

$\bold{(k)}$ log$_a$x = $\frac{log_ax}{log_bx}$ ; x > 0, a,b > 0, a, b $\ne$ 1

$\bold{(l)}$ log$_a$b.log$_b$c.log$_c$d = log$_a$d; a, b, c, d > 0, $\ne$ 1

$\bold{(m)}$ $a^{log_ax}$ = x, a > 0, a $\ne$ 1

$\bold{(n)}$ $a^{log_bc}$ = $c^{log_ba};$ a, b, c > 0, b $\ne$ 1

$\bold{(o)}$ log$_a$x < log$_a$y $\iff$ x = $\begin{cases} x < y &\text{if } a > 1 \\ x > y &\text{if } 0 < a < 1 \end{cases}$

$\bold{(p)}$ log$_a$x = log$_a$y $\Rightarrow$ x = y; x, y > 0; a > 0, a $\ne$ 1

$\bold{(q)}$ $e^{lna^x} = a^x$

$\bold{(r)}$ log$_{10}$2 = 0.3010, log$_{10}$3 = 0.4771, ln2 = 0.693, ln10 = 2.303

$\bold{(s)}$ If a > 1 then log$_a$x < p $\Rightarrow$ 0 < x < $a^p$

$\bold{(t)}$ If a > 1 then log$_a$x > p $\Rightarrow$ x > $a^p$

$\bold{(u)}$ If 0 < a < 1 then log$_a$x < p $\Rightarrow$ x > $a^p$

$\bold{(v)}$ If 0 < a < 1 then log$_a$x > p $\Rightarrow$ 0 < x < $a^p$

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