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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

Sounds are produced when vibrating objects create pressure waves in some medium such as air. When these variations in pressure reach the human eardrum, it causes the eardrum to vibrate in a similar manner and the ear detects sound. The intensity of sound is measured as power per unit area. The threshold for hearing (minimum sound detectable by a young, healthy ear) is defined to be \(I_{0}=10^{-12} \mathrm{~W} / \mathrm{m}^{2}\) (watts per square meter). The sound level \(L,\) or "loudness" of sound, is measured in decibels \((\mathrm{dB})\) as \(L=10 \log \left(\frac{I}{I_{0}}\right),\) where \(I\) is the intensity of the given sound. Use this formula for exercises. a. Find the sound level of a jet plane taking off if its intensity is \(10^{15}\) times the intensity of \(I_{0}\). b. Find the sound level of the noise from city traffic if its intensity is \(10^{9}\) times \(I_{0}\). c. How many times more intense is the sound of a jet plane taking off than noise from city traffic?

Explain why the domain of \(f(x)=x^{2}+k\) must be restricted to find an inverse function.

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?

\(F(C)=\frac{9}{5} C+32\) gives the temperature in degrees Fahrenheit as a function of the temperature \(C\) in degrees Celsius. Find an equation for \(C(F)\) and interpret its meaning in the context of this problem.
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