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Chapter 3: Problem 19

Solve for \(x\). $$e^{x}=2$$

Short Answer

Expert verified
The solution to the equation \( e^{x} = 2 \) is \( x = ln(2) \) or approximately \( x = 0.69314 \).

Step by step solution

01

Apply natural logarithm

Given the equation \( e^{x} = 2 \), which means 2 is equal to \( e \) raised to the power of \( x \). Apply a natural logarithm (ln) to both sides of the equation to be able to remove the exponential of the \( e \). So the equation becomes \( ln(e^{x}) = ln(2) \). The natural logarithm and \( e \) are inverses of each other, and thus they cancel each other out when applied to the same quantity.
02

Simplify equation

Anywhere you see \( ln(e^{x}) \), it simplifies to \( x \) due to the inverse relation mentioned in Step 1. Therefore, the equation now simplifies to \( x = ln(2) \).
03

Calculate natural logarithm of 2

To find the precise value of \( x \), calculate \( ln(2) \) using a calculator or any other method. The value of \( ln(2) \) approximates to 0.69314.

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Most popular questions from this chapter

The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

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The number \(N\) of trees of a given species per acre is approximated by the model \(N=68\left(10^{-0.04 x}\right), 5 \leq x \leq 40,\) where \(x\) is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when \(N=21\).
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