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Chapter 3: Problem 11

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log _{2} \frac{1}{128} $$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: \(\log _{2} \frac{1}{128} = -7\).

Step by step solution

01

Write the logarithmic equation

First, let's write the logarithmic equation corresponding to the given expression. If we let \(x = \log _{2} \frac{1}{128}\), then the equation can be written as: \[2^{x} = \frac{1}{128}\]
02

Convert the fraction to the power of the base

Next, we need to convert the fraction \(\frac{1}{128}\) to the power of the base (2). Since \(128 = 2^{7}\), we can write the fraction as: \[\frac{1}{128} = \frac{1}{2^{7}}\] Now, we can express this as a power of 2 by using the inverse power property: \[\frac{1}{2^{7}} = 2^{-7}\]
03

Solve for x

Now we have our equation in the form: \[2^{x} = 2^{-7}\] Since the bases are the same, the powers must be equal, so we have: \[x = -7\] Thus, the value of the given expression is: \[\log _{2} \frac{1}{128} = -7\]

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Most popular questions from this chapter

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=-6+7 \ln x\)

For each of the functions \(f\); (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part ( \(c\) ) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I .\) (Recall that \(I\) is the function defined by \(I(x)=x .)\) \(f(x)=5 e^{9 x}\)

Find a number \(w\) such that $$ \frac{4-\ln w}{2-5 \ln w}=3.6 $$

Show that \(\sinh x \approx x\) if \(x\) is close to 0 [The definition of \(\sinh\) was given before Problem 60 in Section 3.5.]

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=\log _{5 x} 6 $$
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