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Chapter 4: Problem 77

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

Short Answer

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

Step by step solution

01

Apply the property of logarithms

The expression \(e^{\ln 5 x^{2}}\) represents \(e\) raised to the logarithm base \(e\) of \(5x^2\). Based on the property of logs that states \(e^{ln a} = a\), the expression can be simplified to \(5x^2\).

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Key Concepts

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

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.

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Most popular questions from this chapter

Explain how to use your calculator to find \(\log _{14} 283\).

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

The function \(P(t)=145 e^{-0.022 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. \([\mathrm{TRACE}]\) along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{V x y^{3}}{z^{3}}\right) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x-\frac{1}{3} \ln y $$
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