The
logarithm base b of
a number x is the power to
which b must be raised in order to equal x. This
is written logb x.
For example, log2 8 equals
3 since 23 = 8.
Logarithm Rules:
Algebra rules used when working with logarithm.
For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.
Definitions:
1. loga x = N means that aN = x.
2. log x means log10 x. All loga rules apply for log. When a logarithm is written without a base it means common logarithm.
3. ln x means loge x, where e is about 2.718. All loga rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
Note: ln x is sometimes written Ln x or LN x.
Rules:
1. Inverse properties: loga ax = x and a(loga x) = x
2. Product: loga (xy) = loga x + loga y
3. Quotient:
4. Power: loga (xp) = p loga x
5.Change of base formula:
Common Logarithm:
The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 102 = 100.
Natural Logarithm:
The logarithm base e of a number. That is, the power of e necessary to equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is 2.0794415... since e2.0794415... = 8.
Change of Base Formula:
A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.
Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.
Change of base formula:
Example 1:
Example 2:
(note that 
)
Example 3:
Another examples:
Example 1:
Given: log8(5) = b. Express log4(10) in terms of b.
Solution:
Use log rule of product:
log4(10) = log4(2) + log4(5)
log4(2) = log4(41/2) = 1/2
Use change of base formula to write: log4(5) = log8(5) / log8(4) = b / (2/3) , since log8(4) = 2/3
log4(10) = log4(2) + log4(5) = (1 + 3b) / 2
Example 2:
Simplify without calculator: log6(216) + [ log(42) - log(6) ] / log(49)
Solution:
log6(216) + [ log(42) - log(6) ] / log(49)
= log6(63) + log(42/6) / log(72)
= 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2
Example 3:
Simplify without calculator: ((3-1 - 9-1) / 6)1/3
Solution:
((3-1 - 9-1) / 6)1/3
= ((1/3 - 1/9) / 6)1/3
= ((6 / 27) / 6)1/3 = 1/3
Video:
Logarithm Rules:
Algebra rules used when working with logarithm.
For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.
Definitions:
1. loga x = N means that aN = x.
2. log x means log10 x. All loga rules apply for log. When a logarithm is written without a base it means common logarithm.
3. ln x means loge x, where e is about 2.718. All loga rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
Note: ln x is sometimes written Ln x or LN x.
Rules:
1. Inverse properties: loga ax = x and a(loga x) = x
2. Product: loga (xy) = loga x + loga y
3. Quotient:
4. Power: loga (xp) = p loga x
5.Change of base formula:
Common Logarithm:
The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 102 = 100.
Natural Logarithm:
The logarithm base e of a number. That is, the power of e necessary to equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is 2.0794415... since e2.0794415... = 8.
Change of Base Formula:
A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.
Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.
Change of base formula:
Example 1:
Example 2:
(note that )Example 3:
Another examples:
Example 1:
Given: log8(5) = b. Express log4(10) in terms of b.
Solution:
Use log rule of product:
log4(10) = log4(2) + log4(5)
log4(2) = log4(41/2) = 1/2
Use change of base formula to write: log4(5) = log8(5) / log8(4) = b / (2/3) , since log8(4) = 2/3
log4(10) = log4(2) + log4(5) = (1 + 3b) / 2
Example 2:
Simplify without calculator: log6(216) + [ log(42) - log(6) ] / log(49)
Solution:
log6(216) + [ log(42) - log(6) ] / log(49)
= log6(63) + log(42/6) / log(72)
= 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2
Example 3:
Simplify without calculator: ((3-1 - 9-1) / 6)1/3
Solution:
((3-1 - 9-1) / 6)1/3
= ((1/3 - 1/9) / 6)1/3
= ((6 / 27) / 6)1/3 = 1/3
Video:
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