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Chapter 12: Problem 8

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{9}\left(\frac{9}{x}\right)$$

Short Answer

Expert verified
The expanded expression is \(1 - \log _{9}(x)\)

Step by step solution

01

Apply the division Rule of Logarithms

The division rule of logarithms, states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Therefore we can rewrite \(\log _{9}\left(\frac{9}{x}\right)\) as \(\log _{9}(9) - \log _{9}(x)\)
02

Evaluate and Simplify expressions

We know that \(\log _{9}(9) = 1\), because 9 raised to the first power will yield 9. And we rewrite this expression as \(1 - \log _{9}(x)\)
03

Final Answer

The expanded and simplified expression as required in the task is \(1 - \log _{9}(x)\)

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Most popular questions from this chapter

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Graph the function in a \([0,500,50]\) by \([27,30,1]\) viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the eye increases?

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

$$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

Describe the power rule for logarithms and give an example.

What is an exponential equation?
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