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Chapter 4: Problem 55

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+4)$$

Short Answer

Expert verified
The domain of the function \(f(x)=\log _{5}(x+4)\) is \(x > -4\).

Step by step solution

01

Understand the constraints of a logarithm

Firstly, it is important to understand that the argument of a logarithm has to be greater than 0 (i.e. positive). This is because the logarithm of a negative number or zero is undefined. In our function \(f(x)=\log _{5}(x+4)\), that means that \(x+4\) must be positive.
02

Solve for the domain

Knowing that \(x+4\) must be positive or, in other words, greater than 0, we can now solve for the values of \(x\) that fulfills that requirement. To do this, we can solve the inequality \(x+4 > 0\). Subtracting 4 from each side of the inequality gives us \(x > -4\).
03

Finalize the answer

The solution to the inequality \(x > -4\) tells us that the domain of \(f(x)=\log _{5}(x+4)\) is all real numbers greater than -4. This represents all the x-values which will result in real outputs (y-values) for our function.

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Most popular questions from this chapter

The function \(P(t)=145 e^{-0.022 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. \([\mathrm{TRACE}]\) along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Explain how to use your calculator to find \(\log _{14} 283\).

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

Which one of the following is true? a. \(\frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7\) b. \(\log _{b}\left(x^{3}+y^{3}\right)=3 \log _{b} x+3 \log _{b} y\) c. \(\log _{b}(x y)^{5}=\left(\log _{b} x+\log _{b} y\right)^{5}\) d. \(\ln \sqrt{2}=\frac{\ln 2}{2}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (10,000 x) $$
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