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Chapter 3: Problem 30

evaluate each expression $$\log _{6} \sqrt{6}$$

Short Answer

Expert verified
The value of the expression \(\log _{6} \sqrt{6}\) is \(1/2\).

Step by step solution

01

Write the square root in exponent form

Rewrite the square root of 6 as 6 raised to the power of 1/2: \(\log _{6} 6^{1/2}\). The square root of a number is equivalent to raising that number to the power of 1/2.
02

Apply the logarithm power rule

Apply the power rule of logarithms, which states that \(\log_b a^n = n\log_b a\), to bring down the exponent: \(1/2 \cdot \log_{6} 6\). This rule allows us to move the exponent on the number inside the logarithm to the front as a multiplier.
03

Evaluate the simple log expression

The expression \(\log_{6} 6\) is equal to 1, because any number \(\log_a a\) is equal to 1, since any number raised to the power of 1 gives you the same number. Thus, our final equation is \(1/2 \cdot 1\).
04

Compute the result

Now, continue by calculating the multiplication operation to get the final result. So, \(1/2 \cdot 1 = 1/2\).

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Key Concepts

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

Logarithm Power Rule
The logarithm power rule is essential when handling expressions involving exponents within a logarithm. It states that when you have a log expression like \( \log_b a^n \), you can simplify it by moving the exponent \( n \) in front of the logarithm. This transforms the expression into \( n \cdot \log_b a \).

For example, in the problem \( \log_6 6^{1/2} \), the exponent \( 1/2 \) can be brought outside, simplifying the expression to \( 1/2 \cdot \log_6 6 \). This rule makes calculations easier and helps in understanding the relationships between logs and exponents.

Key points:
  • Logarithm power rule helps simplify expressions.
  • Allows you to break down complex logarithmic forms.
  • Crucial for solving logarithmic equations efficiently.
Exponents
Exponents are a powerful mathematical tool used to express repeated multiplication of a number by itself. In simple terms, if you have a number \( a \) raised to the exponent \( n \), it means \( a \) multiplied by itself \( n \) times. This is written as \( a^n \).

When it comes to square roots, such as \( \sqrt{6} \), they can be expressed using exponents. Specifically, the square root of a number is the same as raising the number to the power of \( 1/2 \). Therefore, \( \sqrt{6} \) can be rewritten as \( 6^{1/2} \).

Important aspects:
  • Exponents simplify repeated multiplication.
  • They help convert roots into logarithm-friendly formats.
  • Understanding their properties aids in learning advanced math concepts.
Square Roots
Square roots are a fundamental aspect of mathematics, used to find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since \( 3 \times 3 = 9 \). This is written as \( \sqrt{9} = 3 \).

Expressing square roots in terms of exponents is crucial when dealing with logarithmic equations like \( \log_b \sqrt{6} \). Here, the square root \( \sqrt{6} \) is expressed as \( 6^{1/2} \). This conversion is particularly useful as it allows the use of rules like the logarithm power rule, making the calculation process more straightforward.

Highlights:
  • Square roots identify a number which squares to the original.
  • Converting to exponents aids in solving log equations.
  • Understanding root properties is vital for algebra and calculus.

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Most popular questions from this chapter

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A\), in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

Consider the quadratic function $$f(x)=-4 x^{2}-16 x+3$$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. (Section \(2.2,\) Example 4)

By 2019 , nearly 1 dollar out of every 5 dollars spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product (GDP) going toward health care from 2007 through 2014 , with a projection for 2019.(GRAPH CAN'T COPY). The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2008 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.6 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.
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