/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Evaluate each expression without... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 30

Evaluate each expression without using a calculator. $$\log _{6} \sqrt{6}$$

Short Answer

Expert verified
The value of \( \log _{6} \sqrt{6} \) is \( 0.5 \)

Step by step solution

01

Identify the problem

The objective is to evaluate \( \log _{6} \sqrt{6} \). In this context, the square root of 6 is equivalent to raising it to the power (exponent) of 0.5.
02

Apply the Logarithm rule

The value of a logarithm is the exponent you have to raise the base to get the argument of the log. In other words, in \( \log _{a} b = c \), \( a^c = b \). Therefore, we must obtain the exponent of the base which gives the argument i.e., \( 0.5 \).
03

Simplify Result

The expression \( \log _{6} \sqrt{6} = \log _{6} 6^{0.5} \) thereby simplifies to \( 0.5 \) because 6 raised to the power of 0.5 equals to square root of 6, which gives us the argument of the log. Hence, the value of the log expression is \( 0.5 \).

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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 \(b>1\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

The formula \(A=22.9 e^{0.0183 t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2005 a. What was the population of Texas in \(2005 ?\) b. When will the population of Texas reach 27 million?

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.
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