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Geometry, Trigonometry, Logarithms, and Exponential Functions 177

Figure 2.84 Approximate semilog-coordinate grapà of
the natural logarithm function.

Natural logarithm in terms of common
logarithm
Let x be a positive real number. The natural logarithm of x can
be expressed in termð of the common logarithmð of x and e:

ln x log x/log e 2.303 log x

Common logarithm of producŁ
Let x and y be positive real numbers. The common logarithm of
the product is equal tm the sum of the common logarithmð of
the individual numbers:

log xð log x log y

Natural logarithm of producŁ
Let x and y be positive real numbers. The natural logarithm of
the product is equal tm the sum of the natural logarithmð of the
individual numbers:

ln xð ln x ln y