Logarithm Rules, Tables, Formulas and Shortcuts

Logarithm Solved Examples - Page 3
Logarithm Important Questions - Page 4
Logarithm Video Lecture - Page 5

Logarithm, in mathematics is the exponent or power to which a stated number called base , is raised to yield a specific number. For example on the expression $10^{2}\, =\, 100$, the Logarithm of 100 to the base 10 is 2. This is written as $Log_{10}\, 100\, =\, 2$ Logarithms were originally invented to help simplify the arithmetical processes of multiplication, division, expansion to a power and extraction of a 'root', but they are now a days used for variety of purposes in pure and applied mathematics.

If for a positive real number (a ≠ 1) , $a^{m}\, =\, b$, then the index m is called the Logarithm of b to the base a.
We write this as: $Log_{a}b\, m$
Log begins the abbreviation of the word ‘Logarithm’. Thus $a^{m}\, b\, \leftrightarrow \, log_{a}b\, =\, m$
Where $a^{m}$ = b is called the exponential form and $Log_{a}b\, =\, m$ is called the Logarithmic form.

Exponential Form
$3^{5}\, =\, 243$
$2^{4}\, =\, 16$
$3^{0}\, =\, 1$
$8^{\frac{1}{3}}\, =\, 2$

Logarithmic Form

$log_{3}243\, =\, 5$
$log_{2}16\, =\, 4$
$log_{3}1\, =\, 0$
$log_{8}2\, =\, \frac{1}{3}$

Logarithm Shortcut Method and Formulas

• Product Formula: The Logarithm of the product of two numbers is equal to the sum of their Logarithms.
i.e. $log_{a}\, (mn)\, log_{a}m\, +\, log_{a}n$
Generalisation: In general, we have
$log_{a}\, (m,n,p,q....)\, =\, log_{a}m\, +\,log_{a}n\, +\,log_{a}p\, +\,log_{a}q\, +\,...$
• Quotient formula: The Logarithm of the quotient of two numbers is equal of their Logarithm.
i.e. $log_{a}\left ( \frac{m}{n} \right )\, =\, log^{a}m\, -\, log_{a}n$. Where a, m, n are positive and a ≠ 1
• Power formula: The Logarithm of a number raised to a power is equal to the power multiplied by Logarithm of the number.
i.e. $log_{a}\, (m^{n})\, =\, n\log_{a}m$. Where a, m are positive and a ≠ 1
• Base changing formula: $log_{n}^{m}\, =\, \frac{log_{a}m}{log_{a}n}\, So,\, log_{n}m\, =\, \frac{log\, m}{log\, n}$ Where m, n, a are positive and n ≠ 1, a ≠ 1.
• Reciprocal Relation: $log_{b}a\, \times \, log_{a}b\, =\, 1$, where a, b are positive and not equal to 1
• $log_{b}a\,=\frac{1}{log_{a}b}$
• $_{a}\textrm{log}_{a}x\, =\,x$, where a and x are positive and a ≠ 1
• If a > 1 and x > 1, then $log_{a}$ x > 0.
• If 0 < a < 1 and 0 < x < 1, then $log_{a}$ x > 0.
• If 0 < a < 1 and x > 1, then then $log_{a}$ x > 0.
• if a > 1 and 0 < x < 1, then then $log_{a}$ x < 0.
• Logarithm of 1 to any base is equal to zero. i.e. $log_{a}$ 1 = 0, where a > 0, a ≠ 1
• Logarithm of any number to the same base is 1. i.e. $log_{a}$ a = 1, where a > 0, a ≠ 1
• Common logarithms: There are two base of logarithms that are extensively used these days. One is base e(e=2.71828 approx) and the other is base 10. The logarithms to base 10 are called the common logarithms.

$log_{10}10\, =\, 1,\, since\, 10^{1}\, =\, 10.$
$log_{10}100\, =\, 2,\, since\, 10^{2}\, =\, 100.$
$log_{10}10000\, =\, 4,\, since\, 10^{4}\, =\, 10000.$
$log_{10}0.01\, =\, -2,\, since\, 10^{-2}\, =\, 0.01.$
$log_{10}0.001\, =\, -3,\, since\, 10^{-3}\, =\, 0.001.$
AND $log_{10}1\, =\, 0,\, since\, 10^{0}\, =\, 1.$

Logarithm Solved Questions

Example 1: $log_{\frac{3}{2}}3.375$
Solution:$log_{\frac{3}{2}}3.375\,= \,x\,\Rightarrow\,\left ( \frac{3}{3} \right )^{x}\,=\,3.375$
$(1.5)^{x}\, =\, (1.5)^{3}\Rightarrow x\, =\, 3$

Example 2: If $x\,=\,log_{2a}a,\, y\, =\, log_{3a}2a\, and\, z\, =\, log_{4a}3a$ find yz(2 - x).
Solution: yz(2 - x) = 2yz - xyz = $2log_{4a}\, 2a\, -\, log_{4a}a$
$=\, log_{4a}\left ( \frac{4a^{2}}{a} \right )\, =\, 1$

Example 3: $\frac{logx}{l+m-2m}\,=\, \frac{logy}{m+n-2l}\,=\, \frac{logz}{n+l-2m}\,,find\, x^{2}y^{2}z^{2}$
Solution:
Each is equal to k
$\Rightarrow$ log x = k(l + m - 2n),
log y = k(m + n - 2l), log z = k(n + l - 2m),
$\Rightarrow$ log xyz = k(0) $\Rightarrow$ xyz = $e^{0}$ = 1 = $x^{2}y^{2}z^{2}$ = 1

Example 4: If log $\frac{x+y}{5}\, =\, \frac{1}{2}(log x + log y),\, then\, \frac{x}{y}+\frac{y}{x}=$
Solution:
$log\left ( \frac{x + y}{5} \right )\, =\, \, \frac{1}{2}\left [ log\, x + log\, y \right ]$
x + y = $\sqrt[5]{xy}\Rightarrow x^{2}\, +\, y^{2}\, =\, 23xy$
$\frac{x}{y}\, \frac{y}{x}\, =\, 23$

Example 5: If log (x + y) = log $\left ( \frac{3x-3y}{2} \right )$, then log x - log y =
Solution:
x + y = $\frac{3x\, -\, 3y}{2}\Rightarrow x\, =\, 5y\Rightarrow \frac{x}{y}\, =\, 5$
log x - log y = log 5

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