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Chapter 7: Problem 57

Find the logarithm by applying the definition of logarithm $$x=\log _{2} 32$$

Short Answer

Expert verified
\(x = 5\)

Step by step solution

01

Understand the Definition of a Logarithm

The logarithm of a number is the power to which the base must be raised to produce that number. In other words, if we have \(x = \log_{b}(y)\), it means that \(b^x = y\).
02

Apply the Definition to the Given Problem

We need to find \(x\) such that \(2^x = 32\). This means we are looking for the exponent \(x\) to which the number 2 must be raised to get 32.
03

Solve for \(x\)

Since 32 is a power of 2 (specifically, \(2^5\)), we can equate \(2^x\) to \(2^5\) because \(32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5\). Therefore, \(x = 5\).

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Key Concepts

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

Logarithm Calculation
Logarithm calculation often stirs confusion, but it's simply a way to unravel the power or exponent that a given base must be raised to, to arrive at a specific value. Let's consider our example, where you are tasked to find the value of x in the equation x = \(\log_{2}{32}\). This is asking, \

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Most popular questions from this chapter

Suppose that \(y\) increases exponentially with \(x\) and that \(z\) is directly proportional to the square of \(x\). Sketch the graph of each type of function. In what ways are the two graphs similar to one another? What major graphical difference would allow you to tell which graph is which if they were not marked?

Evaluate the power of \(10 .\) Then show that the logarithm of the answer is cqual to the original exponent of 10 $$10^{3.5}$$

Test your knowledge of the definition of logarithm Write in logarithmic form: \(m=r^{k}\)

Solve the exponential equation algebraically, using logarithms. $$0.8^{0.4 x}=2001$$

Suppose that \(y\) varies directly with \(x\) and that \(z\) increases linearly with \(x .\) Explain why any direct-variation function is a linear function but a linear function is not necessarily a direct-variation function.
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