Introduction for simplify logarithms:
In arithmetic, the logarithms of a number to be a identified base is the power or exponent to which the base have to be raised in order to generate that number. For example, the logarithms of 10000 to the base 100 is 4, because 4 is the power to the which ten must be raised to produce 10000: 104 = 10000, so log1010000 = 4
Algebra is widely used in day to day activities watch out for my forthcoming posts on algebra 2 math problem solver and free algebra 2 homework help. I am sure they will be helpful.
Simplify Logarithms Rules:
Product rule: If n, A and B are positive numbers and x ?1, then
Logn(AB) = lognA +lognB
Quotient rule: If A, B and n is positive numbers and n ? 1, then,
log n(`A/B` ) = log nA –logn B
Power rule: If n and A are positive numbers, n ? 1 and m is a real number, then
LognAB =BlognA
Change of base rule: If A, B and n are positive numbers and A ? 1, x ? 1, then
log AB = lognA * logBn
Reciprocal rule: If A and B be the positive numbers other than 1, then
logAB =`1/(log_BA) `
Examples for Simplify Logarithms:
Example 1:
Simplify : 56log25+56log26 =56 log2 (6x)
Solution:
logarithmic function56log25+56log26 =56 log2 (6x)
56 log2(5*6) = 56log2(6x)
56log2(30) =56 log2(6x) by logarithm rule
Equate both sides, s the base are same 2, 6x = 30
x =30/6
x = 5
The answer is x = 5.
Example 2:
Simplify : 44log8512-44log88
Solution:
44log8512 - 44log88 = 44[log8(512/8)] { by logarithms rule}
= 44log883
= 3*44 { log88 = 1 }
= 132
The answer is 72.
Example 3:
Simplify logarithms : 12 log289
Solution:
convert the logarithmic equation
= 12log28 { by logarithms rule}
= 12log22 3
= 12*3log22
= 36
The answer is 36.
Example 4:
Simplify : (i) 67 log3 27 - 67log3 9
(ii)56 log5 25 +56 log5 5
Solution:
(i) Since expression is a sum of two logarithms as well as the bases are equal, we can apply the product rule
(i) 67 log3 27 -48log3 729 = 67[log 3 (27/9)]
= 67 [ log3 (32)]
= 67 log3 32
= 67*2 log33
= 67* 2
= 134
(ii) 56log525+56log5(5) = 56log5 (25*5)
= 56 log 5(625)
= 56log 5(5^4)
=56log5(54)
= 4*56log55 = 224
Example 5:
Simplify : log6 9x – log6(7x+1)
Solution:
by quotient law, we can write the equations as log6 `((9x) /( 7x+1))` Changing into exponential form, we get
`((9x) / (7x+1))` = 60 = 1
9x = 7x + 1
x = `1/2`
In arithmetic, the logarithms of a number to be a identified base is the power or exponent to which the base have to be raised in order to generate that number. For example, the logarithms of 10000 to the base 100 is 4, because 4 is the power to the which ten must be raised to produce 10000: 104 = 10000, so log1010000 = 4
Algebra is widely used in day to day activities watch out for my forthcoming posts on algebra 2 math problem solver and free algebra 2 homework help. I am sure they will be helpful.
Simplify Logarithms Rules:
Product rule: If n, A and B are positive numbers and x ?1, then
Logn(AB) = lognA +lognB
Quotient rule: If A, B and n is positive numbers and n ? 1, then,
log n(`A/B` ) = log nA –logn B
Power rule: If n and A are positive numbers, n ? 1 and m is a real number, then
LognAB =BlognA
Change of base rule: If A, B and n are positive numbers and A ? 1, x ? 1, then
log AB = lognA * logBn
Reciprocal rule: If A and B be the positive numbers other than 1, then
logAB =`1/(log_BA) `
Examples for Simplify Logarithms:
Example 1:
Simplify : 56log25+56log26 =56 log2 (6x)
Solution:
logarithmic function56log25+56log26 =56 log2 (6x)
56 log2(5*6) = 56log2(6x)
56log2(30) =56 log2(6x) by logarithm rule
Equate both sides, s the base are same 2, 6x = 30
x =30/6
x = 5
The answer is x = 5.
Example 2:
Simplify : 44log8512-44log88
Solution:
44log8512 - 44log88 = 44[log8(512/8)] { by logarithms rule}
= 44log883
= 3*44 { log88 = 1 }
= 132
The answer is 72.
Example 3:
Simplify logarithms : 12 log289
Solution:
convert the logarithmic equation
= 12log28 { by logarithms rule}
= 12log22 3
= 12*3log22
= 36
The answer is 36.
Example 4:
Simplify : (i) 67 log3 27 - 67log3 9
(ii)56 log5 25 +56 log5 5
Solution:
(i) Since expression is a sum of two logarithms as well as the bases are equal, we can apply the product rule
(i) 67 log3 27 -48log3 729 = 67[log 3 (27/9)]
= 67 [ log3 (32)]
= 67 log3 32
= 67*2 log33
= 67* 2
= 134
(ii) 56log525+56log5(5) = 56log5 (25*5)
= 56 log 5(625)
= 56log 5(5^4)
=56log5(54)
= 4*56log55 = 224
Example 5:
Simplify : log6 9x – log6(7x+1)
Solution:
by quotient law, we can write the equations as log6 `((9x) /( 7x+1))` Changing into exponential form, we get
`((9x) / (7x+1))` = 60 = 1
9x = 7x + 1
x = `1/2`
