/*! 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 37 Solve for \(x\) algebraically. ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 37

Solve for \(x\) algebraically. $$e^{\ln (x-1)}=4$$

Short Answer

Expert verified
The solution to the equation is \(x = 5\).

Step by step solution

01

Simplifying the Expression

Use the property of logarithms and exponential functions, which states that \(e^{\ln a} = a\), where \(e\) is the base of the natural logarithm \(ln\). Applying this rule to our equation, we get: \(e^{\ln (x-1)} = x-1\). Therefore, the equation \(e^{\ln (x-1)}=4\) becomes \(x-1 = 4\).
02

Solving the Equation

The equation \(x-1 = 4\) is a simple linear equation and can now be solved for \(x\). Add 1 to both sides of the equation to isolate \(x\) on one side, resulting in: \(x = 4+1\).
03

Calculating the Answer

Finally, calculate the right side of the equation: \(x = 4+1 = 5\).

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Key Concepts

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

Solving Linear Equations
Linear equations are the backbone of algebra, involving expressions that make a straight line when graphed. A linear equation takes the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. To solve a linear equation, the goal is to isolate \(x\) on one side of the equation.

Here is a straightforward strategy to solve linear equations:
  • Identify and isolate the term with the variable.
  • Move constants to the other side by adding or subtracting them.
  • Divide or multiply to get \(x\) alone, as needed.
Let's apply this to the example equation \(x - 1 = 4\). In this case, you add 1 to both sides to isolate \(x\), which results in \(x = 5\). This simple process demonstrates how you manipulate the equation systematically to reveal the variable's value.
Properties of Logarithms
Logarithms serve as the inverse operation to exponentiation, and they come with a variety of useful properties that simplify complex calculations. A key property to remember is that the exponential function \(e^{\ln a} = a\). This is particularly useful in solving logarithmic equations.

Other important properties include:
  • Product Property: \(\log_b (xy) = \log_b x + \log_b y\).
  • Quotient Property: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\).
  • Power Property: \(\log_b (x^c) = c \log_b x\).
Understanding these properties makes it easier to address equations involving logarithms, just like in the exercise where we used \(e^{\ln (x-1)} = x-1\). This relationship was crucial for simplifying and solving the equation efficiently.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a logarithm with the base \(e\), where \(e \approx 2.718\), a fundamental constant in mathematics. The function \(\ln x\) helps in solving equations related to exponential growth and decay, making it incredibly useful in various fields like finance, science, and engineering.

Key characteristics include:
  • Used to "undo" exponentiation with base \(e\).
  • Commonly appears in formulas involving continuous compounding.
  • The graph of \(\ln x\) displays a curve that increases slowly without bound.
In the exercise, we saw \(\ln (x-1)\) as part of the expression \(e^{\ln (x-1)}\). By recognizing that the natural logarithm and the base \(e\) "cancel" each other, we simplified the equation directly to \(x-1\). This demonstrates the power and utility of the natural logarithm in breaking down complex mathematical expressions.

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Most popular questions from this chapter

Use a graphing utility to graph each function. If the function has a horizontal asymptote, state the equation of the horizontal asymptote. $$f(x)=\frac{e^{x}-e^{-x}}{2}$$

Use a calculator to evaluate the exponential function for the given \(x\) -value. Round to the nearest hundredth. $$g(x)=e^{x}, x=2.2$$

Sketch the graph of each function. $$f(x)=\left(\frac{2}{3}\right)^{x}$$

Explain how to use the graph of the first function \(f\) to produce the graph of the second function \(F\). $$f(x)=3^{x}, F(x)=3^{x}+2$$

Evaluate \(A=600\left(1+\frac{0.04}{4}\right)^{4 t}\) for \(t=8 .\) Round to the nearest hundredth. [3.2]
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