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Chapter 4: Problem 4

Write \(\ln (x+4)=6\) in exponential form.

Short Answer

Expert verified
\( \text{ln}(x + 4) = 6 \) is equivalent to \( x = e^6 - 4 \)

Step by step solution

01

- Understand the logarithmic form

Identify the components in the logarithmic equation \(\text{ln}(x + 4) = 6\). Here, the base of the natural logarithm \(\text{ln}\) is \(e\), the natural base.
02

- Rewrite using exponential form

Use the definition of logarithms to rewrite the equation in exponential form. The definition of logarithms states that \( \text{ln}(a) = b \) is equivalent to \( e^b = a \). Applying this to the equation we get \( e^6 = x + 4 \).
03

- Solve for x

Isolate \( x \) by subtracting 4 from both sides of the equation: \[ x = e^6 - 4 \].

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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 is a special type of logarithm where the base is the constant \( e \), approximately equal to 2.71828. The natural logarithm is denoted as \( \ln \). It is commonly used in many areas of mathematics, especially in calculus and exponential growth problems.
When we write \( \ln(x) \), we mean the logarithm base \( e \) of \( x \). For example, \( \ln(e) = 1 \) because \( e^1 = e \). When dealing with equations involving the natural logarithm, it often helps to convert these logarithmic equations into their exponential form.
exponential equations
An exponential equation is an equation in which a variable appears in the exponent. These types of equations often appear in the context of growth or decay processes. For instance, if we have the natural logarithm equation \( \ln(x + 4) = 6 \), this can be converted to an exponential equation using the properties of logarithms.
According to the definition of logarithms, if \( \ln(a) = b \), then it is equivalent to saying \( e^b = a \). Applying this to our given equation, we can rewrite \( \ln(x + 4) = 6 \) as the exponential equation \( e^6 = x + 4 \).
logarithmic functions
Logarithmic functions are the inverses of exponential functions. They help us solve equations where the variable is an exponent. The most common logarithmic functions include the natural logarithm (base \( e \)) and the common logarithm (base 10).
To solve equations using logarithms, it's often useful to convert logarithmic forms to exponential forms. For example, the natural logarithmic function \( \text{ln}(y) = x \) converts to its exponential form \( e^x = y \). This relationship is crucial for solving equations like our example \( \ln(x + 4) = 6 \), which translates to \( e^6 = x + 4 \).
solving for x
Solving for \( x \) is a common goal in algebra, and often requires the application of logarithmic and exponential forms. Once an equation is converted from its logarithmic to exponential form, isolating \( x \) becomes more straightforward.
With our example \( \ln(x + 4) = 6 \), we first convert this to exponential form to get \( e^6 = x + 4 \). The next step is to isolate \( x \). This is done by subtracting 4 from both sides of the equation:
\[ x = e^6 - 4 \]
This result gives us the value of \( x \) in terms of the mathematical constant \( e \).

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Most popular questions from this chapter

Graph the following functions on the window [-3,3,1] by [-1,8,1] and comment on the behavior of the graphs near $$ \begin{array}{l} x=0 \\ \mathrm{Y}_{1}=e^{x} \\ \mathrm{Y}_{2}=1+x+\frac{x^{2}}{2} \\ \mathrm{Y}_{3}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6} \end{array} $$

Explain why the product property of logarithms does not apply to the following statement. $$ \begin{array}{l} \log _{5}(-5)+\log _{5}(-25) \\ \quad=\log _{5}[(-5)(-25)] \\ \quad=\log _{5} 125=3 \end{array} $$

Which functions are exponential? a. \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) b. \(f(x)=1^{x}\) c. \(f(x)=x^{\sqrt{3}}\) d. \(f(x)=(-2)^{x}\) e. \(f(x)=\pi^{x}\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-6 e^{x}-16=0\)

Solve for the indicated variable. \(M=8.8+5.1 \log D\) for \(D\) (used in astronomy)
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