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Chapter 12: Problem 43

Solve. Where appropriate, include approximations to three decimal places. $$ \ln x=1 $$

Short Answer

Expert verified
The solution is \(x \approx 2.718\).

Step by step solution

01

Understand the Natural Logarithm Equation

The given equation is \(\ln x = 1\). This means that we need to find the value of \(x\) for which the natural logarithm (ln) is equal to 1.
02

Exponential Form of Logarithm

The natural logarithm \(\ln x\) is the same as saying \(\log_e x\), where \(e\) is the base of the natural logarithms, approximately equal to 2.718. To remove the natural logarithm, we use the property that if \(\ln x = y\), then \(x = e^y\). Applying this property, \(x = e^1\).
03

Solve for x

Since \(e\) raised to the power of 1 is simply \(e\), we can write \(x = e^1\). Using a calculator or knowing the approximation, we find that \(e\) is approximately 2.718. So, \(x = 2.718\).
04

Approximations to Three Decimal Places

To ensure the answer is to three decimal places, we round the value of \(e\) appropriately. Since \(e\) is already approximately 2.718, no further rounding is needed.

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Key Concepts

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

solve logarithmic equations
Solving logarithmic equations involves finding the unknown variable for which the logarithmic equation holds true. For instance, if you have an equation like \(\begin{array}{c} ln x = 1 \end{array}\), your goal is to solve for \(x\). Here is the step-by-step approach to solve logarithmic equations:
  • Understand the given logarithmic equation.
  • Express the logarithm in its exponential form if necessary
  • Simplify to find the value of the variable.
In our example, the given equation is \(\ln x = 1\). The aim is to find out what \(x\) equals when the natural logarithm of \(x\) is 1. Follow the steps closely to understand how you can solve equations similar to this.
exponential form
The exponential form is crucial when solving natural logarithmic equations. The natural logarithm, \(\ln x\), is based on the number \(e\), which is approximately 2.718. When you have an equation like \(\ln x = y\), it can be converted to the exponential form as \(x = e^y\).For our example, where \(\ln x = 1\):
  • Convert \(\ln x\) to its exponential form by saying \(x = e^1\).
  • This tells us that \(x\) is the exponential of 1, which simplifies the calculation.
Once in exponential form, it is much easier to manage. For example, knowing \(e^1\) is just \(e\), you can simplify this to approximately 2.718, giving you the value of \(x\).
natural logarithm properties
Understanding natural logarithm properties helps efficiently solve related equations. Here are some key properties of natural logarithms:
  • The natural logarithm of 1 is 0: \(\ln 1 = 0\), because \(e^0 = 1\).
  • The natural logarithm of \(e\) itself is 1: \(\ln e = 1\), because \(e^1 = e\).
  • Useful property for solving equations: If \(\ln x = y\), then \(x = e^y\).
In our specific problem, the equation \(\ln x = 1\) uses the property that if \(\ln x = y\), then \(x = e^y\). Applying this, we get \(x = e^1\), which simplifies to \(x \approx 2.718\). Mastering these properties simplifies the process of solving natural logarithmic equations.

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Most popular questions from this chapter

How could you convince someone that $$\log _{a} c \neq \log _{c} a ?$$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\frac{1}{2} \log _{a} x+5 \log _{a} y-2 \log _{a} x$$

Simplify. $$ \log _{2}\left(\log _{2}\left(\log _{4} 256\right)\right) $$

Solve. $$ \log _{2} x=-1 $$

Find the domain of each function. $$f(x)=\frac{x}{(x-2)(x+3)}$$
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