/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 \(5^{x}=e^{3}\)... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 10

\(5^{x}=e^{3}\)

Short Answer

Expert verified
x = \frac{3}{\text{ln}(5)}

Step by step solution

01

Understand the Problem

We need to solve for the variable \(x\) in the equation \(5^{x}=e^{3}\).
02

Take Natural Logarithms on Both Sides

Apply the natural logarithm on both sides to get \(\text{ln}(5^{x}) = \text{ln}(e^{3})\).
03

Use Logarithm Properties

Using the logarithm property \(\text{ln}(a^b) = b \text{ln}(a)\), we can rewrite the equation as \(x \text{ln}(5) = 3\).
04

Solve for x

Isolate \(x\) by dividing both sides by \(\text{ln}(5)\): \(x = \frac{3}{\text{ln}(5)}\).

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Key Concepts

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

natural logarithms
A natural logarithm, denoted as \(\text{ln}(x)\), is a type of logarithm that uses the mathematical constant 'e' (approximately 2.718) as its base. Natural logarithms are used widely in many areas of mathematics and science. In the context of our exercise, transforming the original exponential equation \(5^x = e^3\) into a more solvable form involves taking the natural logarithm of both sides. This process helps us unlock the exponent and directly deal with the variable \(x\). Remember, natural logarithms have unique properties that make them an excellent choice for solving exponential equations.
logarithm properties
To solve exponential equations like \(5^x = e^3\), we utilize properties of logarithms to simplify expressions. One of the most important properties is \(\text{ln}(a^b) = b \text{ln}(a)\). This allows us to move the exponent in front of the logarithm, making the equation linear and easier to solve. For our specific equation, applying this property transforms \(\text{ln}(5^x)\) into \(x \text{ln}(5)\). This step is crucial because it turns a complex exponentiation problem into a simpler multiplication problem, enabling us to efficiently proceed with solving for \(x\). These logarithm properties are essential tools in algebra and higher mathematics.
isolating variables
Isolating variables is a fundamental technique in algebra for solving equations. Once we've applied the logarithm properties, our equation \(x \text{ln}(5) = 3\) is set up for us to isolate \(\text{x}\). We do this by dividing both sides of the equation by \(\text{ln}(5)\). This isolates \(x\) and leaves us with the final solution: \(\text{x} = \frac{3}{\text{ln}(5)}\).
To recap:
  • We simplified the original problem using natural logarithms.
  • We used the property \(\text{ln}(a^b) = b \text{ln}(a)\).
  • We isolated the variable by performing algebraic operations.
This process is instrumental in solving exponential equations and similar mathematical problems.

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Most popular questions from this chapter

\(y=(x+\ln x)^{3}\) where \(x=1\)

PROFIT A manufacturer of digital cameras estimates that when cameras are sold for \(x\) dollars apiece, consumers will buy \(8,000 e^{-0.02 x}\) cameras each week. He also determines that profit is maximized when the selling price \(x\) is \(1.4\) times the cost of producing cach unit. What price maximizes weekly profit? How many units are sold each week at this optimal price?

In Exercises 43 through 46 , find an equation for the tangent line to the given curve at the specified point. 43\. \(y=x \ln x^{2}\) where \(x=1\)

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In Exercises 31 through 38 , determine where the given function is increasing and decreasing and where its graph is concave upward and concave downward. Sketch the graph, showing as many key features as possible (high and low points, points of inflection, asymptotes, intercepts, cusps, vertical tangents). 31\. \(f(x)=e^{x}-e^{-x}\)
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