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Chapter 11: Problem 79

Use a calculator to solve each equation. Round answers to four decimal places. $$e^{x}=7.2$$

Short Answer

Expert verified
1.9741

Step by step solution

01

Understand the equation

The given equation is an exponential equation in the form \(e^{x} = 7.2\). The goal is to solve for \(x\).
02

Apply the natural logarithm

To isolate \(x\), take the natural logarithm (ln) on both sides of the equation. This gives us \(\ln(e^{x}) = \ln(7.2)\).
03

Simplify the equation

Using the property of logarithms that \(\ln(e^{x}) = x\), simplify the left side of the equation to get \(x = \ln(7.2)\).
04

Use a calculator to find the value

Calculate \(\ln(7.2)\) using a calculator. Make sure to round the result to four decimal places.
05

State the final answer

The value of \(x\) after rounding to four decimal places is approximately \(1.9741\).

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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 written as 'ln', is a special kind of logarithm. It's the inverse of the exponential function with base 'e'. The number 'e' is approximately equal to 2.71828 and is irrational.
This means that the natural logarithm of a number is the power to which 'e' must be raised to get that number.
For example, \(\text{ln}(7.2) = x\) means \(e^{x} = 7.2\). When we take the natural logarithm on both sides of the exponential equation \(e^{x} = 7.2\), we use the property that \(\text{ln}(e^{x}) = x\). This simplifies our problem and makes it easier to isolate and solve for 'x'.
Exponential Function
An 'exponential function' is a mathematical function in which an independent variable appears in the exponent. The general form is \(a^{x}\), where 'a' is a positive real number, and 'x' is the exponent.
The exponential function with base 'e', \(e^{x}\), is especially important in mathematics due to its unique properties.
Here are some key properties:
  • \(e^{0} = 1\)
  • \(e^{1} \approx 2.71828\)
  • The function grows extremely fast for positive 'x'.
In our exercise, we have \(e^{x} = 7.2\). Our task is to determine the value of 'x' that makes this equation true. By applying the natural logarithm to both sides, we simplify it to find the exact value of 'x'.
Calculator Usage in Algebra
Using a calculator can make solving algebraic equations much simpler, especially when dealing with natural logarithms and exponential functions.
Here鈥檚 how to use a calculator to find \(\text{ln}(7.2)\):
  • First, make sure your calculator is in 'Scientific' mode to access the 'ln' function.
  • Enter 7.2 and press the 'ln' button.
You should see something like 1.9741.
Make sure to round this answer to four decimal places as required.
Using calculators correctly can save time and reduces the risk of manual errors. It's also a great habit to cross-check your work.

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Most popular questions from this chapter

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$\log _{7}\left(x^{2}+6 x+8\right)-\log _{7}(x+2)=\log _{7}(3)$$

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$2^{x}=\frac{1}{3}$$

Determine whether each equation is true or false. $$ \log _{2}\left(\frac{5}{2}\right)=\log _{2}(5)-1 $$

Determine whether each equation is true or false. $$ \ln \left(4^{2}\right)=(\ln (4))^{2} $$

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$10^{x-2}=6$$
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