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Chapter 5: Problem 19

Find the exact value of the expression without using your GDC. $$\log _{2} 64$$

Short Answer

Expert verified
The exact value is 6.

Step by step solution

01

Understanding the Logarithmic Expression

The given expression is \(\log_{2} 64\). This means that we are looking for the power to which the base 2 must be raised to get 64.
02

Express 64 as a Power of 2

Next, observe that 64 can be expressed as a power of 2. Since \(2^6 = 64\), we can rewrite 64 as \(2^6\).
03

Rewrite the Logarithmic Equation

Using the expression from the previous step, \(\log_{2} 64\) can be rewritten as \(\log_{2} 2^6\).
04

Apply the Power Rule of Logarithms

Use the power rule of logarithms: \(\log_{b} b^x = x\) for any base \(b\) and exponent \(x\). Therefore, \(\log_{2} 2^6 = 6\).
05

Conclusion

The exact value of the expression \(\log_{2} 64\) is \(6\).

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Key Concepts

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

Base of Logarithms
Logarithms are mathematical expressions that help us determine the power to which a given number, known as the base, must be raised to produce a specific value. In the expression \(\log_b x\), \(b\) is the base. The base is essential to understanding the logarithmic function. For instance, in \(\log_{2} 64\), the number 2 is our base. It tells us that we are trying to find the exponent needed for 2 to become 64. Understanding this base is key to solving for logarithms, as it sets the foundation for the relationship between the numbers involved.
Exponentials
Exponential numbers are numbers raised to a power, often written in the form \(b^x\), where \(b\) is the base and \(x\) is the exponent. Expanding our understanding of logarithms requires knowing how to express numbers as exponentials. This is vital because logarithms essentially "undo" this exponentiation process. For example, expressing 64 as \(2^6\) (since \(2^6 = 64\)) is an exponential expression. This helps us use the power rule of logarithms effectively, simplifying the process of finding the logarithm's value. In essence, exponentials and logarithms are inverse operations: exponentiation "builds" the number, while logarithms "break it down."
Power Rule for Logarithms
The power rule is a handy tool in logarithms, adding simplicity when dealing with exponents within a logarithmic function. The power rule states that \(\log_b b^x = x\). This means when the base of a logarithm matches the base of the power inside the log, you can simplify directly to the exponent. Returning to the expression \(\log_{2} 64\), which we rewritten as \(\log_{2} 2^6\), the power rule allows us to resolve this to 6 almost instantly. Utilizing the power rule makes solving such problems quicker and more straightforward, providing a neat shortcut to arrive at the answer.

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Most popular questions from this chapter

The population of a city triples every 25 years. At time \(t=0,\) the population is \(100000 .\) Write a function for the population \(P(t)\) as a function of \(t .\) What is the population after: a) 50 years b) 70 years c) 100 years?

Solve for \(x\) in the logarithmic equation. Give exact answers and be sure to check for extraneous solutions. $$\log _{2}\left(x^{2}+8\right)=\log _{2} x+\log _{2} 6$$

Write each expression as the logarithm of a single quantity. $$\log x-\log y-\log z$$

Use the change of base formula to evaluate \(f(20)\) $$f(x)=\log _{5} x$$

Find the exact value of the expression without using your GDC. $$\log _{3}\left(3^{5}\right)$$
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