/*! 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 73 Let \(\log _{b} 2=A\) and \(\log... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 73

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} \sqrt{\frac{2}{27}}$$

Short Answer

Expert verified
The expression \(\log _{b} \sqrt{\frac{2}{27}}\) converted in terms of \(A\) and \(C\) is \(0.5 ( A - 3C )\)

Step by step solution

01

- Represent Square Roots as Fractional Exponent

Rewrite \(\sqrt{\frac{2}{27}}\) as \(\left(\frac{2}{27}\right)^{0.5}\). So, the expression becomes: \(\log _{b} \left(\frac{2}{27}\right)^{0.5}\)
02

- Apply the Power Rule of Logarithms

The power rule of logarithms states that \(\log_b M^n=n \log_b M\). Accordingly, the expression from step 1 becomes: \(0.5 \log _{b} \left(\frac{2}{27}\right)\)
03

- Apply Logarithm Rule for Fractions

Using the rule \(\log_b \frac{M}{N} = \log_b M - \log_b N\), the quantity within the log from step 2 can be separated: \(0.5 \left( \log _{b} 2 - \log _{b} 27 \right)\)
04

- Break Down 27 into Base 3

Since we want to write the expression in terms of \(\log _{b} 2\) and \(\log _{b} 3\), rewrite 27 as \(3^3\). Now, the expression becomes: \(0.5 \left( \log _{b} 2 - \log _{b} (3^3) \right)\)
05

- Apply Power Rule & Write in terms of A and C

Using the power rule of logarithms to the expression \(\log _{b} (3^3)\), it comes out to be \(3 \log _{b} 3\). Substituting \(\log _{b} 2 = A\) and \(\log _{b} 3 = C\), the final expression will be: \(0.5 ( A - 3C )\)

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Most popular questions from this chapter

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _______.

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?

Solve: $$\sqrt{2 x-1}-\sqrt{x-1}=1$$

What is a logarithmic equation?

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a nuptured eardrum. Use the formula to solve Exercises. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?
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