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Logarithm is an important topic that needs to be prepared well for the Quantitative Aptitude section of exams in India. It requires practicing a lot of questions within time. Logarithmic formulas make it easy to solve questions easily in competitive examinations. The following article covers the concepts, formulas, and rules that a learner needs to know before approaching the questions.

Logarithmic function is inverse to the exponential function. A logarithm to the base b is the power to which b must be raised to produce a given number. For example, is equal to the power to which 2 must be raised in order to produce 8. Clearly, 2^3 = 8 so = 3. In general, for b > 0 and b not equal to 1.

Logarithm Formulas

There are several logarithm formulas, but some of the most common ones are:

• Logarithmic identity:
log(b^x) = x * log(b)
• Product rule:
log(b, xy) = log(b, x) + log(b, y)
• Quotient rule:
log(b, x/y) = log(b, x) – log(b, y)
• Power rule:
log(b, x^p) = p * log(b, x)
• Change of base formula:
log(b, x) = log(a, x) / log(a, b)

Where:

• b is the base of the logarithm
• x and y are the arguments of the logarithm
• p is a constant
• a is a different base, usually chosen to be 10 or e.

Sample Questions on Logarithm with Solutions:

Question 1: Find the value of x in equation given 8x+1 – 8x-1 = 63

Solution:

Take 8x-1 common from the eq.
It reduce to
8x-1(82 – 1) = 63
8x-1 = 1
Hence, x – 1 = 0
x = 1

Question 2: Find the value of x for the eq. given log0.25x = 16

Solution:

log0.25x = 16
It can be write as
x = (0.25)16
x = (1/4)16
x = 4-16

Question 3: Solve the equation log121728 x log96561

Solution:

It can be written as:
log12(123) x log9(94)
= 3log1212 x 4log99
= 3 x 4 = 12

Question 4: Solve for x : logx3 + logx9 + logx27 + logx81 = 10

Solution:

It can be write as:
logx(3 x 9 x 27 x 81) = 10
logx(31 x 32 x 33 x 34) = 10
logx(310) = 10
10 logx3 = 10
then, x = 3

Question 5: If log(a + 3 ) + log(a – 3) = 1 ,then a=?

Solution:

log10((a + 3)(a – 3))=1
log10(a2 – 9) = 1
(a2 – 9) = 10
a2 = 19
a = √19

Question 6: Solve 1/logab(abcd) + 1/logbc(abcd) + 1/logcd(abcd) + 1/logda(abcd)

Solution:

=logabcd(ab) + logabcd(bc) + logabcd(cd) + logabcd(da)
=logabcd(ab * bc * cd * da)
=logabcd(abcd)2
=2 logabcd(abcd)
=2

Question 7: If xyz = 10 , then solve log(xn yn / zn) + log(yn zn / xn) + log(zn xn / yn)

Solution:

log(xn yn / zn * yn zn / xn * zn xn / yn)
= log xn yn zn
= log(xyz)n
= log10 10n
= n

Question 8: Find (121/10)x = 3

Solution:

Apply logarithm on both sides
log(121/10)(121/10)x = log(121/10)3
x = (log 3) / (log 121 – log 10)
x = (log 3) / (2 log 11 – 1)

Question 9: Solve log(2x2 + 17)= log (x – 3)2

Solution:

log(2x2 + 17)= log (x2 – 6x + 9)
2x2 + 17 = x2 – 6x + 9
x2 + 6x + 8 = 0
x2 + 4x + 2x + 8 = 0
x(x + 4) + 2(x + 4) = 0
(x + 4)(x + 2)=0
x= -4,-2

Question 10: log2(33 – 3x)= 10log(5 – x). Solve for x.

Solution:

Put x = 0
log2(33 – 1)= 10log(5)
log232 = 5
5 log2 2 = 5
5 = 5
LHS = RHS

Also Practice:

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