Evaluate or simplify each expression without using a calculator. $e^{\ln 5x^2}$

Verified step by step guidance

1. Recognize that the expression is \(e^{(\ln 5x^2)}\), which involves the exponential function and the natural logarithm function.
2. Recall the property of logarithms and exponentials: \(e^{\ln a} = a\) for any positive \(a\).
3. Apply this property to simplify \(e^{(\ln 5x^2)}\) directly to \$5x^2$.
4. Note that this simplification holds as long as the argument inside the logarithm, \$5x^2$, is positive, which means \(x \neq 0\).
5. Therefore, the simplified form of the expression is \$5x^2$.

Key Concepts

Properties of Exponents

Understanding how to manipulate expressions with exponents is essential. For example, the rule e^(a + b) = e^a * e^b allows breaking down complex exponentials, and e^(ln x) simplifies directly to x when x is positive.

Natural Logarithm (ln) and Its Inverse Relationship with Exponentials

The natural logarithm ln(x) is the inverse function of the exponential function e^x. This means e^(ln y) = y for y > 0, which helps simplify expressions like e^(ln 5x^2) directly to 5x^2.

Simplifying Algebraic Expressions

After applying logarithmic and exponential properties, simplifying the resulting algebraic expression is necessary. This includes understanding how to handle powers, coefficients, and variables, such as recognizing that (x^2) remains as is unless further simplification is possible.