Logarithms – properties (Changing logarithms from one base to another) – pg.8/10
pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
7. Changing logarithms from one base to another
For this demonstration we are going to use the notations in [1] and [2], pg.2.
[18] (shown in [1], pg.2)
[19] (shown in [2], pg.2)
Let’s assume that:
* [20] or as a logarithmic expression:
[21]
* [22] or as a logarithmic expression:
[23]
By replacing [20] and [22] into [18] we have:
(see [13] in § 12 Raising to the power an exponential expression in Exponents (Powers))
[24]
From [24] we have and
[25]
By replacing in [25]:
* from [19]
* from [21]
* from [23]
the expression [25] will become::
[26]
If (common logarithm), our expression becomes:
[27]
If (natural logarithm), our expression becomes:
[28]
The general expression for changing a logarithm from one base to another is:
[29]

The general expression for changing a logarithm from one base to common logarithm is:

The general expression for changing a logarithm from one base to natural logarithm is:

Example:
Let’s consider the logarithmic expression:
Let’s change this logarithm into logarithms that have different base:
1. Let’s consider the change into a logarithm of base :
2. Let’s consider the change into a logarithm of base :
3. Let’s consider the change into a logarithm of base :
pg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
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