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Chapter 4: Problem 106

Evaluate the expressions. $$ \log _{2}\left[\log _{1 / 2}\left(\frac{1}{4}\right)\right] $$

Short Answer

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Step by step solution

01

- Evaluate Inner Logarithm

First, simplify the innermost logarithm, \ \log _{1 / 2}(\frac{1}{4}). Recall that \(\frac{1}{4} = \left(\frac{1}{2}\right)^2 \). Use the logarithm property \(\log_b(b^x) = x \). Therefore, \(\log _{1 / 2}(\frac{1}{4}) = \log _{1 / 2}((\frac{1}{2})^2) = 2 \).
02

- Substitute and Simplify Outer Logarithm

Now substitute the simplified value of the inner logarithm into the outer logarithm. We have: \ \log _{2}(2). \ Using the property \(\log_b(b) = 1 \, we find that \log _{2}(2) = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Logarithms are the inverse operations of exponentiation. They help us solve equations involving exponential terms. Here are some key properties of logarithms that will help you work with them more effectively:

- **Product Property**: \(\, \log_b(MN) = \log_b(M) + \log_b(N) \)
- **Quotient Property**: \(\, \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \)
- **Power Property**: \(\, \log_b(M^k) = k\log_b(M) \)
- **Change of Base Formula**: \(\, \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

Understanding these properties allows you to break down and simplify complex logarithmic expressions. They are especially useful when dealing with multiple logarithms or changing bases.
Inner and Outer Logarithms
When evaluating expressions with nested logarithms, you must start with the innermost logarithm first. In the expression \(\, \log_{2}[\log_{\frac{1}{2}}(\frac{1}{4})] \), we first focus on the inner term \(\, \log_{\frac{1}{2}}(\frac{1}{4}) \).

Recall that \(\frac{1}{4} = (\frac{1}{2})^2\). Using the property \(\, \log_b(b^x) = x \), we can simplify the inner logarithm. Thus, \(\, \log_{\frac{1}{2}}(\frac{1}{4}) = \log_{\frac{1}{2}}((\frac{1}{2})^2) = 2 \).

Once the inner logarithm is simplified, it can be substituted back into the outer logarithm. This step-by-step approach ensures you simplify correctly and avoid errors.
Change of Base Formula
Sometimes you need to evaluate logarithms with a base that is not convenient or is unfamiliar. This is where the change of base formula comes in handy. The formula is:

\( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

For example, if you need to find \(\, \log_{5}(125) \) and do not have a calculator that supports base 5, you can use base 10 (common logarithm) or base e (natural logarithm):

\(

\log_{5}(125) = \frac{\log_{10}(125)}{\log_{10}(5)} \)

Using this formula, the logarithm becomes something you can calculate easily with standard functions on a calculator.
Evaluating Logarithms
Evaluating logarithms involves breaking them down step-by-step. Let's revisit the given exercise:

First, simplify the inner logarithm:

\(\, \log_{\frac{1}{2}}(\frac{1}{4}) \).

Represent \(\frac{1}{4} \) as an exponent of \(\frac{1}{2} \):
\(\frac{1}{4} = ((\frac{1}{2})^2) \).

Use \(\, \log_b(b^x) = x \):

\(\, \log_{\frac{1}{2}}((\frac{1}{2})^2) = 2 \).

Now substitute back into the expression:

\(\, \log_{2}(2) \).

Using the identity \(\, \log_{b}(b) = 1 \), we have \(\, \log_{2}(2) = 1 \).

Always follow the order: simplify inner logarithms first, then substitute into the outer logarithm, and use logarithmic properties to simplify further.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(80=320 e^{-0.5 t}\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}=-7 e^{x}\)

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{5} z=3-\log _{5}(z-20)\)

a. Graph \(f(x)=\ln x\) and \(g(x)=(x-1)-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}-\frac{(x-1)^{4}}{4}\) on the viewing window [-2,4,1] by [-5,2,1] . How do the graphs compare on the interval (0,2) ? b. Use function \(g\) to approximate \(\ln 1.5\). Round to 4 decimal places.
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