Problem 77 - College Algebra

Question

Use inverse properties of logarithms to simplify each expression. $$e^{\ln 5 x^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(5x^2\).

Step 1: Identify the Expression

We evaluate or simplify:
Use inverse properties of logarithms to simplify each expression.
$$e^{\ln 5 x^{2}}$$

Step 2: Apply Logarithm Properties

Key properties: \(\log_a(a^x) = x\), \(\log_a(1) = 0\), \(e^{\ln x} = x\), \(\ln(e^x) = x\).

Step 3: Result

The simplified expression is \(5x^2\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form, making them easier to work with and understand. When using inverse properties of logarithms, the goal is to identify parts of the expression that can be reduced using known mathematical properties. In the given exercise, we have the expression \(e^{\ln 5x^{2}}\). This expression can be simplified by using the inverse property of logarithms, which states that \(e^{\ln a} = a\). This means when an exponential function contains a logarithm with the same base, they essentially cancel each other out, leaving only the argument of the logarithm.

• Identify the base of the exponent and the logarithm. Here both are base \(e\).
• Apply the property \(e^{\ln a} = a\).
• Simplify the expression to obtain \(5x^2\).

As a result, the expression is simplified to \(5x^2\), making calculations much more straightforward.

Exponential Functions

Exponential functions involve mathematical expressions where numbers or variables are raised to a power. Here, the base is typically a constant like \(e\), which is an irrational number approximately equal to 2.71828. In mathematics, exponential functions are expressed in the form \(y = b^x\), where \(b\) is the base, and \(x\) is the exponent. The function \(e^{\ln 5x^2}\) falls under this category, where \(e\) is the base and \(\ln 5x^2\) is the exponent.

• Understanding that when you see \(e\) and a natural log \(\ln\) together, they are inverse functions.
• Recognizing how exponential functions grow and change rapidly, which is why they are useful in many fields such as biology, finance, and physics.
• Remembering that inverse properties simplify calculations by removing the logarithmic part.

These functions play a critical role in simplifying expressions by utilizing their properties effectively, as done in the original exercise.

Properties of Logarithms

Logarithms transform multiplication into addition, division into subtraction, powers into multiplications, and roots into divisions. One of the fundamental properties is the inverse property, which is crucial for simplifying expressions.The exercise utilizes the property \(e^{\ln a} = a\). This signifies the inverse relationship between exponential and logarithmic functions when they share the same base. The base of the natural logarithm is \(e\), which perfectly aligns with the base of the exponential function given by \(e^{\ln 5x^2}\).

• The inverse property cancels out the effect of the logarithm by the exponential function, leaving us with \(5x^2\).
• Other key properties include the product rule, \(\ln(ab) = \ln a + \ln b\), and the power rule, \(\ln(a^b) = b\ln a\).
• Enhancing problem-solving by breaking down complex equations into simpler forms.

These properties make logarithms a powerful tool, allowing us to tackle otherwise tedious calculations efficiently.