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
Answer: 3
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).
Answer: 1
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}
Answer: 1
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}=
Answer: 23
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 =
Answer: Log 5
Solution:
x + y = \frac{3x\, -\, 3y}{2}\Rightarrow x\, =\, 5y\Rightarrow \frac{x}{y}\, =\, 5
log x - log y = log 5

 

Logarithm Questions from Previous Year Exams.

  • Logarithm Aptitude

 

Logarithm Online Study Material - Video Tutorial

 

 

 

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