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Chapter 4: Problem 17

Find the indicated value of the logarithmic functions. $$\log _{2}(64)$$

Short Answer

Expert verified
The value of \(\text{log}_{2}(64)\) is 6.

Step by step solution

01

Identify the Logarithmic Function

The problem asks to find the value of \(\text{log}_{2}(64)\). This means we need to determine what power of 2 equals 64.
02

Convert to Exponential Form

Rewrite the logarithmic equation \(\text{log}_{2}(64)\) as an exponential equation: \(2^x = 64\).
03

Solve the Exponential Equation

Determine the value of \(x\) for which \(2^x = 64\). Notice that \(64 = 2^6\). Therefore, \(x = 6\).
04

Conclude the Result

Since \(\text{log}_{2}(64) = 6\), the indicated value of the logarithmic function is 6.

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Key Concepts

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

Logarithms
A logarithm answers the question: For a given number, what power must another fixed number (the base) be raised to in order to produce that given number? For example, in \(\text{log}_{2}(64)\), we are trying to find out how many times we need to multiply 2 by itself to get 64. This 'how many times' is the exponent. So, if we say \(\text{log}_{b}(a) = c\), it means \({b^c = a}\). We usually write this as \(\text{log}_{b}(a) = c\). With this in mind, let鈥檚 explore core aspects of logarithms through different lenses.
Exponential Equations
Exponential equations involve expressions where variables appear as exponents. They have the form \({b^x = a}\). We often convert logarithmic equations into exponential equations to make them easier to solve. In our example, \(\text{log}_{2}(64)\), we converted it to \({2^x = 64}\). This is a straightforward process that makes solving for the exponent more intuitive. To solve \({2^x = 64}\), we need to recognize patterns or use known exponent rules. Since \({2^6 = 64}\), we see that \({x = 6}\). This type of conversion is crucial in simplifying both the process and understanding of equations.
Base 2 Logarithm
The base of a logarithm is critical to solving the equation correctly. In the case of \(\text{log}_{2}(64)\), the base is 2. This means we are dealing with powers of 2. The base 2 logarithm is particularly common in computer science and information theory because it deals with binary numbers. In general, base 2 logarithms help us understand how many times a number must be divided by 2 to get to 1. This is useful in algorithms and computational complexity. So, \(\text{log}_{2}(64) = 6\) tells us that 2 raised to the power of 6 equals 64.
Logarithmic to Exponential Conversion
The conversion from logarithmic form to exponential form is a key skill in solving logarithmic equations. To convert \(\text{log}_{b}(a) = c\) to exponential form, you rewrite it as \({b^c = a}\). This conversion allows us to deal with the equation in a more familiar form. For our problem \(\text{log}_{2}(64) = x\), we write it in exponential form as \({2^x = 64}\). By doing this, we leverage our understanding of exponents to solve the equation. Recognizing that \({2^6 = 64}\) immediately gives us that \({x = 6}\). This method enhances comprehension and enables efficient problem-solving.

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Most popular questions from this chapter

Solve each problem. When needed, use 365 days per year and 30 days per month. Population Growth The function \(P=2.4 e^{0.03 t}\) models the size of the population of a small country, where \(P\) is in millions of people in the year \(2000+t\) a. What was the population in \(2000 ?\) b. Use the formula to estimate the population to the nearest tenth of a million in 2020 .

Evaluate \(\left(2 \times 10^{-9}\right)^{3}\left(5 \times 10^{3}\right)^{2}\) without a calculator. Write the answer in scientific notation.

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Solve each problem. When needed, use 365 days per year and 30 days per month. Present Value Compounding Daily A credit union pays \(6.5 \%\) annual interest compounded daily. What deposit today (present value) would amount to \(\$ 3000\) in 5 years and 4 months? Hint Solve the compound interest formula for \(P\).

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