The natural logarithm of one-half, denoted as ln(1/2), is indeed equivalent to the negative of the natural logarithm of two, -ln(2). This equivalence arises from the properties of logarithms, specifically the logarithm of a quotient. Understanding this relationship is crucial for simplifying expressions and solving equations involving logarithms.
Understanding the Logarithm of a Quotient
The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This can be expressed mathematically as:
log<sub>b</sub>(x/y) = log<sub>b</sub>(x) - log<sub>b</sub>(y)
where:
- b is the base of the logarithm (in our case, the natural logarithm, which has a base of e)
- x is the numerator
- y is the denominator
Applying the Logarithm of a Quotient Rule to ln(1/2)
Let's apply this rule to our specific case of ln(1/2):
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Identify the numerator and denominator: In the expression ln(1/2), the numerator is 1, and the denominator is 2.
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Apply the logarithm of a quotient rule:
ln(1/2) = ln(1) - ln(2) -
Simplify using the logarithm of 1:
ln(1) = 0 (since any number raised to the power of 0 equals 1).
Therefore, ln(1/2) = 0 - ln(2) = -ln(2).
Visualizing the Relationship
The relationship between ln(1/2) and -ln(2) can also be visualized graphically. The graph of y = ln(x) is a curve that passes through the point (1, 0). Since ln(1/2) is the y-coordinate of the point where x = 1/2, and -ln(2) is the negative of the y-coordinate where x = 2, they are symmetric about the y-axis.
Practical Applications
The equivalence between ln(1/2) and -ln(2) is frequently used in calculus, physics, and engineering. For example, it is essential for solving differential equations, understanding radioactive decay, and analyzing logarithmic scales.
Conclusion
The equivalence of ln(1/2) and -ln(2) is a direct consequence of the logarithm of a quotient rule. Understanding this relationship is crucial for simplifying expressions, solving equations, and interpreting logarithmic functions. The visualization of the relationship on the graph of y = ln(x) further enhances this understanding. This knowledge is essential for tackling various problems in mathematics and its applications.
