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Chapter 4: Problem 110

a. Evaluate \(\log _{3} 3+\log _{3} 27\) b. Evaluate \(\log _{3}(3 \cdot 27)\) c. How do the values of the expressions in parts (a) and (b) compare?

Short Answer

Expert verified
\(\log _{3} 3 + \log _{3} 27\) and \(\log_{3}(3 \cdot 27)\) both evaluate to 4.

Step by step solution

01

Evaluate \(\log _{3} 3\)

To evaluate \(\log _{3} 3\), remember that \(\log_b b = 1\). Therefore, \(\log _{3} 3 = 1\).
02

Evaluate \(\log _{3} 27\)

Rewrite 27 as a power of 3. Since \(\27 = 3^3\), we have: \(\log _{3} 27 = \log_{3} (3^3) \). Using the property \(\log_b(b^x) = x\), \(\log _{3} 27 = 3\).
03

Add the logarithms from parts (a)

Combine the results of steps 1 and 2: \(\log _{3} 3 + \log _{3} 27 = 1 + 3 = 4\).
04

Evaluate \(\log _{3}(3 \cdot 27)\)

Use the property of logarithms: \(\log_b(x \cdot y) = \log_b x + \log_b y\). So, \(\log _{3}(3 \cdot 27) = \log_{3} 3 + \log_{3} 27\). From steps 1 and 2, this is \(\1 + 3 = 4\).
05

Compare the results

The value of \(\log _{3} 3 + \log_{3} 27\) matches the value of \(\log_{3} (3 \cdot 27)\). Both are equal to 4.

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Key Concepts

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

Logarithm Properties
Understanding the core properties of logarithms is essential when working with logarithmic functions. Here are some critical properties you need to know:
  • Product Property: \[ \text{log}_b(x \times y) = \text{log}_b(x) + \text{log}_b(y) \] This property states that the logarithm of a product is equal to the sum of the logarithms of its factors.

  • Quotient Property: \[ \text{log}_b\bigg(\frac{x}{y}\bigg) = \text{log}_b(x) - \text{log}_b(y) \] This property notes that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.

  • Power Property: \[ \text{log}_b(x^y) = y \times \text{log}_b(x) \] The logarithm of a number raised to a power is the exponent multiplied by the logarithm of the base number.
Knowing these properties and how to apply them can simplify complex logarithmic expressions.
Logarithmic Equations
Logarithmic equations involve finding unknown values that make the equation true when using logarithms. Here are some steps to solve them:
  • Isolate the logarithm: Ensure the logarithmic term is by itself on one side of the equation.

  • Rewrite exponential form: Convert the logarithmic equation into its equivalent exponential form. For example, \[ \text{log}_b(a) = c \] becomes \[ b^c = a \]

  • Solve for the variable: Proceed by solving the equation for the unknown variable.
Sometimes, you may need to use properties of logarithms to combine or expand logarithmic terms before solving.
Base of Logarithm
The base of a logarithm is a fundamental component of logarithmic functions. Here's what you need to know:
  • Definition: A logarithm \[ \text{log}_b(a) \] is the power to which the base \[ b \] must be raised to get \[ a \]. In exponential form, it's written as \[ b^c = a \] where \[ c = \text{log}_b(a) \].

  • Common bases: The two most commonly used bases are 10 and e (where \[ e \] is approximately 2.718). \[ \text{log}_{10} \] is called the common logarithm, often written simply as \[ \text{log} \]. \[ \text{log}_e \] is known as the natural logarithm and is denoted as \[ \text{ln} \].

  • Base Conversion: You can convert logarithms from one base to another using the change of base formula: \[ \text{log}_b(a) = \frac{\text{log}_c(a)}{\text{log}_c(b)} \]. This is particularly useful when you need to compute logarithms with less common bases using a calculator.
Understanding how to work with different bases can help in solving more advanced logarithmic problems.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{8}\left(\frac{1}{w}\right)=-\log _{8} w $$

Technetium- \(99\left({ }^{99 \mathrm{~m}} \mathrm{Tc}\right)\) is a radionuclide used widely in nuclear medicine. \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is combined with another substance that is readily absorbed by a targeted body organ. Then, special cameras sensitive to the gamma rays emitted by the technetium are used to record pictures of the organ. Suppose that a technician prepares a sample of \(^{99 \mathrm{~m}}\) Tc-pyrophosphate to image the heart of a patient suspected of having had a mild heart attack. a. At noon, the patient is given \(10 \mathrm{mCi}\) (millicuries) of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\). If the half-life of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is \(6 \mathrm{hr}\), write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) after \(t\) hours. b. At what time will the level of radioactivity reach \(3 \mathrm{mCi} ?\) Round to 1 decimal place.

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{4}(3 d)+\log _{4} 1=\log _{4}(3 d) $$

Prove the power property of logarithms: \(\log _{b} x^{p}=p \log _{b} x\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{2} w-3=-\log _{2}(w+2)\)
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