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Chapter 6: Problem 17

Change each exponential statement to an equivalent statement involving a logarithm. $$ e^{x}=8 $$

Short Answer

Expert verified
ln(8) = x

Step by step solution

01

Identify the Exponential Form

In the given exercise, the exponential equation is \(e^{x}=8\).
02

Understand Exponential and Logarithmic Relationships

Recall that the exponential expression \(b^y = x\) can be written in logarithmic form as \(\text{log}_b(x) = y\).
03

Rewrite the Given Equation Using Logarithms

Here, the base \(b\) is \(e\), the exponent \(y\) is \(x\), and the result \(x\) is \(8\). Applying the logarithmic relationship, write \(\text{log}_e(8) = x\).
04

Simplify the Logarithmic Expression

The natural logarithm with base \(e\) is denoted as \(\text{ln}\). Therefore, \(\text{log}_e(8)\) can be simplified to \(\text{ln}(8)\). Hence, \(\text{ln}(8) = x\).

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Key Concepts

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

Exponential Form
Exponential form involves expressions where a number is raised to a power. For example, in the exercise, we are given the equation \( e^x = 8 \). Here, \( e \) is the base, \( x \) is the exponent, and \( 8 \) is the result. Exponential equations like this one can describe many real-world phenomena, including compound interest and population growth.

Understanding the components:
  • The base \( e \) is a constant approximately equal to 2.71828.
  • The exponent \( x \) indicates how many times the base is multiplied by itself.
  • The result, in this case, is \( 8 \).
Often, converting exponential forms to logarithmic forms can simplify the problem and help in solving for the unknown exponent.
Logarithmic Form
A logarithmic form provides an alternate way to express exponential equations. The general form \( b^y = x \) in logarithmic terms becomes \( \text{log}_b(x) = y \). Converting the given exponential equation \( e^x = 8 \) to logarithmic form, we identify:
  • The base \( b \) is \( e \).
  • The result \( x \) is \( 8 \).
  • The exponent \( y \) is \( x \).
In our example, this translates to \( \text{log}_e(8) = x \). The logarithm \( \text{log}_b(x) \) tells us the exponent to which the base \( b \) must be raised to produce a given number \( x \). This relationship is critical in solving equations involving exponential growth or decay.
Natural Logarithm
The natural logarithm, denoted as \( \text{ln} \), is a specific type of logarithm with the base \( e \). When you see an expression like \( \text{log}_e(x) \), you can simplify it to \( \text{ln}(x) \). This simplification is useful because the natural logarithm appears frequently in calculus and higher mathematics.

For example:
  • Given \( e^x = 8 \), converting this to logarithmic form gives us \( \text{log}_e(8) = x \).
  • This can be simplified to \( \text{ln}(8) = x \).
Hence, solving the equation requires finding the natural logarithm of \( 8 \), which tells us that \( x = \text{ln}(8) \). Natural logarithms are often used in problems involving exponential growth, physics, and in many differential equations.
Base e
The number \( e \), approximately 2.71828, is known as Euler's number. It is the base of natural logarithms and has a special place in mathematics due to its unique properties, especially in calculus.

Some key points about \( e \):
  • It is an irrational number, meaning it cannot be written as a simple fraction.
  • It serves as the base for natural logarithms.
  • It appears naturally in many contexts such as in continuous compound interest, in complex numbers, and in the fundamental theorem of calculus.
When working with exponential and logarithmic equations, understanding \( e \) and its properties can greatly simplify the solving process, just as we've seen in the conversion of \( e^x = 8 \) to \( \text{ln}(8) = x \).

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Most popular questions from this chapter

Use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three decimal places. \(\log _{1 / 3} 71\)

Graph each function using a graphing utility and the Change-of-Base Formula. \(y=\log _{4}(x-3)\)

Solve each equation. $$ \log _{3}(3 x-2)=2 $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve the inequality: \(\frac{x+1}{x-2} \geq 1\).

Find the remainder \(R\) when \(f(x)=6 x^{3}+3 x^{2}+2 x-11\) is divided by \(g(x)=x-1 .\) Is \(g\) a factor of \(f ?\)
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