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Chapter 1: Problem 60

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. $$\log _{3} 30$$

Short Answer

Expert verified
Answer: The value of $\log _{3} 30$ is approximately 3.1072.

Step by step solution

01

Identify the given logarithm

The given logarithm is: $$\log _{3} 30$$
02

Change of base formula

We will use the change of base formula to rewrite the logarithm in terms of the natural logarithm. The change of base formula is given by: $$\log _{b} a = \frac{\log _{c} a}{\log _{c} b}$$ where \(a\) is the argument, \(b\) is the base, and \(c\) is the new base. In our case, \(a=30\), \(b=3\), and \(c=e\) (natural logarithm).
03

Apply the change of base formula

Using the change of base formula, we have: $$\log _{3} 30 = \frac{\ln 30}{\ln 3}$$
04

Use a calculator to find the value of the logarithm

Input the expression \(\frac{\ln 30}{\ln 3}\) into a calculator to evaluate the value of the logarithm and round the result to four decimal places: $$\frac{\ln 30}{\ln 3} \approx 3.1072$$ So, the value of the given logarithm is approximately: $$\log _{3} 30 \approx 3.1072$$

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Most popular questions from this chapter

Without using a graphing utility, sketch the graph of \(y=\log _{2} x .\) Then on the same set of axes, sketch the graphs of \(y=\log _{2}(x-1), y=\log _{2} x^{2}\) \(y=\left(\log _{2} x\right)^{2},\) and \(y=\log _{2} x+1\)

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