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A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors. In symbols, logo(XY)=logo(x)+logo(y). (x y) = log b (x) + log b
A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors. In symbols, logo(XY)=logo(x)+logo(y). (x y) = log b (x) + log b
When we take the logarithm of both sides of ELN(XY)=ELN(x)+LN(y), we obtain LN(ELN(XY))=LN(ELN(x)+LN(y)). The logarithms and exponential cancel each other out (equation (4)), giving our product rule for logarithms, LN(XY)=LN(x)+LN(y).
0:24 5:03 Suggested clip Sum of logarithms with same base | Logarithms | Algebra II | Khan YouTubeStart of suggested client of suggested clip Sum of logarithms with same base | Logarithms | Algebra II | Khan
0:00 0:51 Suggested clip Learn how to condense natural logs separated by subtraction YouTubeStart of suggested client of suggested clip Learn how to condense natural logs separated by subtraction
The real deal is that you cannot take the log (or LN) of a number that actually has units, i.e., before the log (or LN) is applied, the unit must be dimensionless. You may be familiar with the concept of making quantities that otherwise have units, unitless, as being referred to as activities in chemistry.
”Since logarithm is the inverse of the exponent, it MUST work for units also.” Nope. When you take the log of a number (base e, base 10, base 43.538, base) the units cancel and the log is a unitless quantity. If you then use a power function you can not recover the units.

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