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Chapter 4: Problem 9

\(4 \ln x=8\)

Short Answer

Expert verified
The solution is \( x = e^2 \).

Step by step solution

01

Understand the Equation

The given equation is \(4 \ln x = 8\). First, identify that it's an equation involving a logarithm.
02

Isolate the Natural Logarithm

Divide both sides of the equation by 4 to isolate \(\ln x\): \[ \frac{4 \ln x}{4} = \frac{8}{4} \] This simplifies to: \ln x = 2.
03

Exponentiate Both Sides

Since \(\ln x = 2\), exponentiate both sides using the base of the natural logarithm, which is e. Thus, \[ e^{\ln x} = e^2 \] Recall that \( e^{\ln x} = x \). Therefore, \[ x = e^2. \]

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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 denoted as \(\text{ln}\), is a logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. It has unique properties that make it especially useful in calculus and solving certain types of equations.

When you see \(\text{ln} x\), it means the power to which e must be raised to obtain x. In other words, if \(\text{ln} x = y\), then \(\text{e}^y = x\).

Some key points to remember about the natural logarithm include:
  • \(\text{ln}(1) = 0\) because \(e^0=1\)
  • \(\text{ln}(e) = 1\) because \(e^1= e\)
  • \(\text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b)\)
  • \(\text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b)\)
  • \(\text{ln}(x^y) = y \times \text{ln}(x)\)

Understanding these properties will help you manipulate and solve equations involving natural logarithms more efficiently.
Exponential Function
The exponential function, denoted as \(e^x\), is one of the most important functions in mathematics, especially in calculus and complex analysis. It is the inverse function of the natural logarithm.

The function \(f(x) = e^x\) has several key characteristics:
  • The rate of growth of \(e^x\) is proportional to its current value.
  • The graph of \(e^x\) is always increasing and never touches the x-axis.
  • It passes through the point (0,1) because \(e^0=1\).
  • The derivative of \(e^x\) with respect to x is \(e^x\).

One of the main reasons the exponential function is so critical is its relationship with the natural logarithm. If \(y = e^x\), then the natural logarithm of y (log base e) brings us back to x: \(\text{ln} y = x\). This inverse property is essential when solving logarithmic equations.
Solving Logarithmic Equations
Solving logarithmic equations often requires the use of natural logarithms and exponential functions. Let鈥檚 take a closer look at solving the equation \(4 \text{ln} x = 8\).

  • Step 1: Understand the Equation
    The given equation involves a natural logarithm, and our goal is to isolate the variable x.
  • Step 2: Isolate the Natural Logarithm
    Divide both sides by 4 to simplify the equation:
    \(\frac{4 \text{ln} x}{4} = \frac{8}{4}\)
    This reduces to \(\text{ln} x = 2\).
  • Step 3: Exponentiate Both Sides
    Exponentiating both sides with base e helps us get rid of the natural logarithm:
    \(e^{\text{ln} x} = e^2\)
    We know that \(e^{\text{ln} x} = x\), so:
    \(\text{x} = e^2\)

Therefore, the solution to the equation is x=\text{e}^2. This method of solving works for many logarithmic equations. Always try to isolate the logarithm first and then exponentiate to solve for the variable.

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Most popular questions from this chapter

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?

COMPOUND INTEREST How quickly will \(\$ 2,000\) grow to \(\$ 5,000\) when invested at an annual interest rate of \(8.5\) if interest is compounded: a. Quarterly b. Continuously

Use a graphing utility to draw the graphs of \(y=\ln \left(1+x^{2}\right)\) and \(y=\frac{1}{x}\) on the same axes. Do these graphs intersect?

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\. \(f(x)=e^{x}-e^{-x}\)

BACTERIAL GROWTH The number of bacteria in a certain culture grows exponentially. If 5,000 bacteria were initially present and 8,000 were present 10 minutes later, how long will it take for the number of bacteria to double?
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