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Chapter 12: Problem 66

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{0.3} 19$$

Short Answer

Expert verified
To round to four decimal places, the result should be -1.4624.

Step by step solution

01

Use change of base formula

Firstly, apply the change of base formula which states that \( \log_{b}a = \dfrac{\log a}{\log b} \). In this problem, the base (b) is 0.3 and the number (a) is 19. Therefore, we substitute these values into the equation, we get: \( \log_{0.3}19 = \dfrac{\log 19}{\log 0.3} \)
02

Calculate using a calculator

Next, use a calculator to evaluate the right side of the equation. Be sure to remember that the calculator uses base 10 by default, so there is no need for any adjustment when inputing in the values.
03

Round off to four decimal places

Finally, round off the final value obtained to four decimal places. This is a standard practice in many scientific and engineering disciplines to ensure the precision and accuracy of the result.

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Most popular questions from this chapter

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can use any positive number other than 1 in the changeof-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\).

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$

Explain how to solve an exponential equation when both sides can be written as a power of the same base.
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