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Chapter 3: Problem 31

Find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) $$\log _{2} \sqrt[4]{8}$$

Short Answer

Expert verified
The exact value of the logarithmic expression \(\log _{2} \sqrt[4]{8}\) is 1/2.

Step by step solution

01

Find the Fourth Root of 8

The fourth root of a number \(x\) is the number which, when raised to the power of 4, will give \(x\) as the result. Here, we have to find the fourth root of 8, which is \(8^{1/4}\). This simplifies to \(8^{1/4} = 2^{3/4}\) = 2^(3/2/2) = \(2^{3/2 * 1/2}\) = \(2^{3/4}\) = \(2^{1/2}\) = \(\sqrt{2}\). So, \(\sqrt[4]{8}\) simplifies to \(\sqrt{2}\) in the base 2 log expression.
02

Compute the Base 2 Logarithm of the Root

Since we have found that \(\sqrt[4]{8}\) is \(\sqrt{2}\), now we write it in our base 2 logarithm to get \(\log _{2} \sqrt{2}\). The logarithm base 2 of \(\sqrt{2}\) gives us the power to which we need to raise 2 to get \(\sqrt{2}\). Here, that power is 1/2. Therefore, \(\log_{2}\sqrt{2}\) is equal to 1/2.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 4 x-\log (12+\sqrt{x})=2$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+1)-\ln (x-2)=\ln x$$

(a) solve for \(P\) and (b) solve for \(t\). $$A=P e^{r t}$$

Use your school's library, the Internet, or some other reference source to write a paper describing John Napier's work with logarithms.
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