Logarithm Problems and Solutions

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.                                                                                                        (Source.Wikipedia)

I like to share this Logarithm Properties with you all through my article.

Logarithms Rules:

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 for of Logarithm Problems and Solutions:

Problem 1: Solve log24+log28=log2 (16x)

Solution:

logarithmic function log23+log28=log2 (16x)

log2(4*8)  = log2(16x)

log2(16x)=log2(32)  by logarithm rules

Equate both sides, s the base are same 2, 16x = 32

Simplification: x = `32/16`

Answer = 2

Problem 2:

Simplify 2log642-2log67

Solution:

2lo642-2log67

=2[log6(`42/7` )] by logarithms rule

Simplification: 2log66

Answer = 1*2=2

Problem 3:

logarithmSolving equation in   2 log286

Solution:

Converting the logarithmic equation

=  2*6log28  by logarithms rule

Simplification: 2*6log223 = 12*3log22                      {   therefore logaa=1}

Answer=36

More practice problem for  logarithms:

I am planning to write more post on Subtract Decimals and How do you Add Decimals. Keep checking my blog.

Problem  1:

Reduce: 13log3 27 + 13log3 729   (ii)15 log5 8 +15  log5 5/1000

Solution:

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

(i) 13log3 27 +13log3 729 =13 [log 3 (27*729)]

=13[ log3 (33*36)]

= 13log3 39 =13* 9 log33 = 13*9=117

(ii)  15 log58+15log5(`5/1000` )  =15 log5 (8*`5/1000` )

=15 log 5(`1/25` )

=15log 5(`1/ 5^2` )

=15 log5(5-2)

= -2*15 log55 = -30*1= -30

Problem  2:

solve: 13log7 98-13log714

Solution:

13log7 98-13log714 =13log 7(`98/14` )

=13 log77

=13*1

= 13

Problem  3:

Solve .log651x – log6(x+8)

Solution: Using quotient law, we can write the equations as log6 (51x / x+8) Changing into exponential form, we get

(51x /x+8) = 60=1

51x = x + 8

50x=8

x =`8/50`