Problem 134 Simplify. $$ \log _{2}\left(... [FREE SOLUTION]

Chapter 12: Problem 134

Simplify. $$ \log _{2}\left(\log _{2}\left(\log _{4} 256\right)\right) $$

Short Answer

Expert verified

Step by step solution

Simplify the innermost logarithm

First, simplify $\log _{4} 256$. To do this, rewrite 256 as $4^4$, since $4^4 = 256$. Therefore: \[ \log_{4} 256 = \log_{4} (4^4) \]\[ \log_{4} (4^4) = 4 \]

Substitute the simplified value into the next logarithm

Now substitute the value obtained from Step 1 into the next logarithmic expression: $\log _{2}(\log _{2}(4))$

Simplify the second logarithm

Next, simplify $\log _{2} 4$. Since $4 = 2^2$, we have:\[ \log_{2} 4 = \log_{2} (2^2) \]\[ \log_{2} (2^2) = 2 \]

Substitute the simplified value into the outer logarithm

Substitute the value from Step 3 into the outer logarithmic expression: $\log _{2} 2$

Simplify the outermost logarithm

Simplify $\log _{2} 2$ directly using the fact that $2 = 2^1$:\[ \log_{2} 2 = \log_{2} (2^1) \]\[ \log_{2} (2^1) = 1 \]

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Key Concepts

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

logarithmic properties

Understanding logarithmic properties is essential for simplifying complex expressions. One key property is the ability to rewrite the argument of a logarithm in exponential form. For example, if you have \ $ \log_{b}(a^{c}) \ $, you can simplify it to \ \ c \cdot \log_{b}(a) \ \.

nested logarithms

Nested logarithms involve taking the logarithm of another logarithmic expression. Decomposing and simplifying from the innermost expression outward ensures clarity. For example, in the expression \ $ \log_{2}\big( \log_{2}\big( \log_{4} 256 \big) \big) \ $, start by simplifying the innermost part \ \ \log_{4} 256 \ \ to 4. Replacing the inner value incrementally steps you forward smoothly.

change of base formula

The change of base formula allows you to compute logarithms with different bases, making complex calculations manageable. It states that \ $ \log_{a}(b) = \frac{ \log_{c}(b) }{ \log_{c}(a) } \ $. For example, converting \ $ \log_{4} 256 \ $ into base 2 gives \ $ \frac{ \log_{2}(256) }{ \log_{2}(4) } = \frac{8}{2} = 4 \ $.

logarithm rules

Logarithm rules simplify expressions quickly. Here are some key rules:
1. The Product Rule: \ $ \log_{b}(MN) = \log_{b}(M)\text{ + } \log_{b}(N) \ $.
2. The Quotient Rule: \ $ \log_{b}(\frac{M}{N}) = \log_{b}(M)\text{ - } \log_{b}(N) \ $.
3. The Power Rule: \ $ \log_{b}(M^{k}) = k \cdot \log_{b}(M) \ $.
Recognizing when to apply these can make simplifications more straightforward.

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91影视

Simplify. $$ \log _{2}\left(\log _{2}\left(\log _{4} 256\right)\right) $$

Short Answer

Step by step solution

Simplify the innermost logarithm

Substitute the simplified value into the next logarithm

Simplify the second logarithm

Substitute the simplified value into the outer logarithm

Simplify the outermost logarithm

Key Concepts

logarithmic properties

nested logarithms

change of base formula

logarithm rules

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