Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:

b. exp(x − y) = exp(x) / exp(y)

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Recall that the exponential function \( \exp(x) \) and the natural logarithm \( \ln(x) \) are inverse functions, meaning \( \exp(\ln(x)) = x \) for \( x > 0 \) and \( \ln(\exp(x)) = x \) for all real \( x \).

Start with the expression \( \exp(x - y) \). Since \( \exp \) and \( \ln \) are inverses, write \( \exp(x - y) = \exp(\ln(\exp(x - y))) \).

Use the property of logarithms that \( \ln(a/b) = \ln(a) - \ln(b) \). To apply this, rewrite \( x - y \) as \( \ln(\exp(x)) - \ln(\exp(y)) \) because \( \ln(\exp(x)) = x \) and \( \ln(\exp(y)) = y \).

Substitute back into the exponential function: \( \exp(x - y) = \exp(\ln(\exp(x)) - \ln(\exp(y))) \). Using the logarithm property, this equals \( \exp(\ln(\exp(x)/\exp(y))) \).

Since \( \exp \) and \( \ln \) are inverse functions, \( \exp(\ln(\exp(x)/\exp(y))) = \exp(x)/\exp(y) \). Thus, we have shown that \( \exp(x - y) = \frac{\exp(x)}{\exp(y)} \).

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Inverse Relationship Between exp(x) and ln(x)

The exponential function exp(x) and the natural logarithm ln(x) are inverse functions, meaning exp(ln(x)) = x for x > 0 and ln(exp(x)) = x for all real x. This relationship allows us to switch between exponential and logarithmic forms to simplify expressions and prove identities.

Properties of the Natural Logarithm ln(x)

The natural logarithm has key properties such as ln(ab) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b) for positive a and b. These properties help break down complex expressions into simpler parts, which is essential when manipulating expressions involving exp(x) through their logarithmic counterparts.

Exponential Function Laws

The exponential function satisfies laws similar to those of powers, including exp(x + y) = exp(x) * exp(y) and exp(0) = 1. Using these laws, one can rewrite expressions like exp(x - y) as exp(x) / exp(y), which is the property to be proven.

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:b. exp(x − y) = exp(x) / exp(y)