Simplify each expression. ln(e^3) = ln(e^(2y)) =
The Correct Answer and Explanation is:
Simplified Expressions
- ln(e3)=3\ln(e^3) = 3
- ln(e2y)=2y\ln(e^{2y}) = 2y
Explanation
To simplify logarithmic expressions such as ln(e3)\ln(e^3) and ln(e2y)\ln(e^{2y}), we rely on fundamental properties of logarithms and exponents, especially those involving the natural logarithm (ln\ln) and the constant ee.
The natural logarithm function ln(x)\ln(x) is the logarithm to the base ee, where ee is an irrational number approximately equal to 2.718. One of the key properties of logarithms is: ln(ex)=x\ln(e^x) = x
This is because the natural logarithm and the exponential function with base ee are inverse operations. When you take the natural log of an exponential expression where the base is ee, the exponent comes down as a simplified result.
Example 1:
ln(e3)\ln(e^3)
Using the identity ln(ex)=x\ln(e^x) = x, we replace the exponent: ln(e3)=3\ln(e^3) = 3
Example 2:
ln(e2y)\ln(e^{2y})
Again, using the same rule: ln(e2y)=2y\ln(e^{2y}) = 2y
This simplification works for any real-valued exponent. Whether the exponent is a number, a variable, or an algebraic expression, the rule still applies: as long as you’re taking the natural logarithm of an exponential expression with base ee, the result is just the exponent.
This property is useful in calculus, algebra, and many applied sciences because it allows complex exponential-logarithmic expressions to be simplified easily, making differentiation, integration, or solving equations much easier.
In summary:
- ln(ex)=x\ln(e^x) = x
- ln(e3)=3\ln(e^3) = 3
- ln(e2y)=2y\ln(e^{2y}) = 2y
These simplifications are direct applications of the inverse relationship between logarithms and exponential functions.
