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Chapter 12: Problem 70

Simplify each expression. $$e^{\ln 7 x^{2}}$$

Short Answer

Expert verified
\(7x^2\)

Step by step solution

01

Identify Exponential and Logarithmic Identity

From the given expression \(e^{\ln 7 x^{2}}\), we can observe that there is an exponential function and a natural logarithm involved. One of the fundamental properties of logarithms states that \(e^{\ln x} = x\). Thus, we can apply this property directly to the given expression.
02

Apply Logarithmic Identity and Simplify

We apply the logarithmic identity \(e^{\ln x} = x\) to simplify the given expression. In this case, \(x\) is 7x^2. So, \(e^{\ln 7 x^{2}} = 7x^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, often represented by "ln", is a specific type of logarithm that uses the mathematical constant \( e \) as its base. This constant \( e \) is approximately equal to 2.71828. The natural logarithm and the exponential function \( e^x \) are inverse operations. This relationship means they effectively cancel each other out when composed with one another.

For example:
  • The natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain the number \( x \).
  • Mathematically, if \( \, y = \ln x \, \), then \( \, e^y = x \, \).

In logarithmic form, natural logarithms simplify expressions by converting multiplication into addition and powers into multiplication, which can be particularly useful in calculus and algebra applications.
Exponential Function
The exponential function is a mathematical function denoted by \( e^x \), where \( e \) is Euler's number, about 2.71828. This function is unique due to its growth rate, which is proportional to its current value. Exponential functions play a crucial role in modeling various phenomena like population growth, radioactive decay, and compound interest.

Key properties of the exponential function:
  • It is always positive, i.e., \( e^x > 0 \) for all \( x \).
  • The derivative of \( e^x \) with respect to \( x \) is \( e^x \) itself, which gives it the unique property of self-similarity in growth.
  • The function \( e^x \) acts as an inverse to the natural logarithm function \( \ln x \).

This inverse relationship means that when you have \( e \) raised to the natural logarithm of a number, it simplifies back to the original number, as seen in the exercise.
Logarithmic Identity
Logarithmic identities are mathematical equations that simplify complex logarithmic expressions. A particularly useful identity is \( e^{\ln x} = x \). This identity exploits the inverse relationship between the exponential function and the natural logarithm. It tells us that raising \( e \) to the power of \( \ln x \) returns \( x \) because these operations are inverses.

In practical use:
  • This identity is vital for solving expressions that involve both \( e \) and \( \ln \), as shown in the simplification of \( e^{\ln 7x^2} \) to \( 7x^2 \).
  • It helps to transform and solve equations that would otherwise be cumbersome or complex.

Understanding and applying these identities is crucial for efficiently managing logarithmic and exponential expressions in mathematics.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

$$\text { Divide and simplify: } \frac{\sqrt[3]{40 x^{2} y^{6}}}{\sqrt[3]{5 x y}}$$

I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

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