Simple Logarithms

In mathematics, the logarithms of a number to a given base is the power or exponent to which the base must be raised in order to produce that number. For example, the logarithm of 100000 to base 10 is 5, because 5 is the power to which ten must be raised to produce 100000: 105 = 100000, so log10100000  = 5. Only positive real number  have real number logarithms; negative and complex numbers have complex logarithms.

I like to share this Simplifying Logarithms with you all through my article.

Simple logarithms are simple step produced by the problem.
Simple Logarithms Rules:

Let us see some of the simple steps that used to solve the logarithims.

Product rule: If a, p and q are positive numbers and a ?1, then

loga(pq) = logap +logaq

Quotient rule: If p, q and a are positive numbers and a ? 1, then,

log a(p/q) = log ap –loga q

Power rule: If a and q are positive numbers, a ? 1 and m is a real number, then

logapq =qlogap

Change of base rule: If p, q and a are positive numbers and p ? 1, a ? 1, then

Log pq = logap* logqa

Reciprocal rule: If p and q is the positive numbers other than 1, then

Log pq =1 / log pq
Examples Simple Logarithms:

Example 1:

Reduce: 22log3 27 + 22log3 729      (ii)75 log5 8 +75  log5 5/1000

Solution:

(i) Since the expressions is a sum of two logarithms and the bases are equal, we can apply the product rule

(i) 22log3 27 +4log3 729 =22 [log 3 (27*729)]

=22[ log3 (33*36)]

= 22log3 39 =22* 9 log33 = 22*9=198

(ii)               75 log58+75log5(`5/1000` )  =75 log5 (`(8*5)/1000` )

=75 log 5(`1/25` )

=75log 5(`1/ 5^2` )

=75 log5(5-2)

= -2*75 log55 = -150*1= -150



My forthcoming post is on prime factorization method, and  formula for conditional probability will give you more understanding about Algebra.


Example 2:

solve: 89log7 98-89log714

Solution:

89log7 98-89log714 =89log 7(`98/14` )

=89 log77 =89*1= 89

Example 3:

Solve .log650x – log6(4x+1)

Solution:

Using quotient law, we can write the equations as log6 (50x / 4x+1) Changing into exponential form, we get

(50x /4x+1) = 60

50x = 4x + 1

46x=1

x=`1/46`