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Logarithm

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Logarithm is an exponent measure. An equation in exponential form can be equivalently written in logarithmic form. For example, we have the exponential statement, 23 = 8. This can be written in logarithmic form as log2 8 = 3. Here, 2 is the base in both the equations and only the positions of 3 and  8 are switched. Equations in one form are often written in another form and solved. As functions, exponential and logarithmic functions are inverses of each other. Corresponding to properties of exponents, we also have properties of logarithm.

Define Logarithm

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For x > 0 and a > 0, and a $\neq$ 1,
y = loga x  <=> x = ay.
In other words, the equations y = loga x and x = ay are equivalent equations.
We can understand y as the power to which a is raised to get x.
y = loga ay.
The function f(x) defined as f(x) = loga x is called the logarithmic function with base a.
The logarithms is defined only for positive numbers. The base of a logarithm is also positive and never equal to 1.As a0 = 1, loga 1 = 0. That is, logarithm of 1 to any base = 0.
Also, log 0 to any base is written as - $\infty$.

Common Logarithm

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Base 10 logarithms are known as common logarithm, as they are easy to manipulate and used widely in calculations.
The base of a logarithm by default is 10. That is, for the expression log x, it is understood that the base is 10.
The common logarithms of powers of 10 are integers.
log 10 = log10 101 = 1
log 100 = log10 102 = 2
log 10,000 = log10 104 = 4
log 1 = log10 100 = 0
log 0.1 = log10 10-1 = -1
log 0.001 = log10 10-3 = -3

In general, using inverse property, log 10x = x  and 10log x = x.
Any number can be expressed as a power of 10. The numerical calculations involving multiplication, division and exponentiation are done taking logarithms and using properties of logarithm. The output of these logarithmic manipulations are then raised to the power of 10 (anti logarithm) to get the required result.
All Scientific calculators have the 'log' function key which is used for evaluating common logarithm.

Natural Logarithm

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The base in natural logarithm is the natural number or Euler's constant 'e'. Natural logarithm is represented by 'ln'.
ln x =  loge x where, x > 0.
y = ln x  <=>  x = ey.
The function f(x) = ln x is the inverse function of the natural exponential function y = ex.
In general, ln ex = x and eln x = x (By Inverse Property)

Properties of Logarithms

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The common rules applicable to logarithms are also known as properties or laws of logarithms. The first three laws equivalent to laws of exponents are used for simplifying exponents.

Product Rule:
Logarithm of a product is equal to the sum of the logarithms of the factors.
logb (xy) = logb x + logb y
log 400 = log 100 + log 4 = 2 + log 4

Quotient Rule:
Logarithm of a quotient is the difference of the logarithms of the two quantities that make the quotient.
logb (x/y) = logb x - logb y
log2 (8/3) = log2 8 - log2 3  = log2 23 - log2 3 = 3 - log2 3.

Power Rule:
logb xm = m logb x
log 27 = log 33 = 3 log 3

Change of Base Rule:
In addition to the above three rules, change of base rule is extensively used in simplifying and evaluating logarithmic expressions. The rule to change the base from b to a is as follows:
logb x = $\frac{\log_{a}x}{\log_{a}b}$

log
2 5 = $\frac{\log _{10}5}{\log _{10}2}$ = $\frac{\log 5}{\log 2}$

Expanding Logarithms

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The product, quotient and power rules are used in expanding logarithms.

Solved Example

Question: Expand the logarithm log2 $\frac{9(x - 2)}{x^{2}}$
Solution:
log2 $\frac{9(x - 2)}{x^{2}}$ = log2 (9(x - 2)) - log2 x2  (Quotient Rule)
                             = log2 9 + log2 (x - 2) - log2 x2     (Product Rule)
                             = log2 32 + log2 (x - 2) - log2 x2                    
                             = 2 log2 3 + log2 (x - 2) - 2 log2 x   (Power Rule)

Adding Logarithms

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Logarithms with same bases are added to form a single logarithm using the product rule reversely.
Examples:
  • log 2 + log 5 = log (5 . 2) = log 10 = log10 101 = 1.
  • 3 ln x + ln 4 = ln x3 + ln 4 = ln (x3 . 4) = ln (4x3)

Subtracting Logarithms

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The quotient rule is also used reversely to condense a difference of logarithms. Let us see with the help of examples:
  • log 16 - 2 log 2 = log 16 - log 22 = log 16 - log 4 = log (16/4) = log 4
  • 3 ln(x) - ln(y) = ln x3 - ln y = ln (x3/y)

How to Solve Logarithms?

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Logarithmic equations are solved using either one-to-one property of logarithmic functions or its inverse property.

Solved Examples

Question 1: Solve log2 (x2 + 5x) = log2 6
Solution:
x2 + 5x = 6                 One - to - One Property of Logarithms
x2 + 5x - 6 = 0
(x + 6) (x - 1) = 0
=> x = - 6 and x = 1.
Both the values satisfy the given equation. Hence, the two solutions are x = - 6, 1.

Question 2: Solve log4 (7x + 2) = 2
Solution:
7x + 2 = 42    (By definition of logarithm)
7x + 2 = 16
7x = 14   
x = 4

Logarithm Practice Problems

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Given below are some of the practice problems in logarithm.

Practice Problems

Question 1: Solve log (10x) + log (x - 3) = 2.
Question 2: Simplify: log4 2 + log4 8
Question 3: Calculate log7 12 using change of base rule.