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Chapter 4: Problem 8

In Exercises \(1-40,\) 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 form of the given logarithmic expression \(\log_{9}(9/x)\) is \(1 - \log_{9}x\).

Step by step solution

01

Expand using quotient property

Firstly, we will use the quotient property of logarithms to expand the logarithmic expression, we have \(\log_{9}(9/x) = \log_{9}9 - \log_{9}x\).
02

Simplify using knowledge of logarithms

Since log base 9 of 9 is simply 1, the first part of the equation simplifies to 1 and thus our expression is now \(1 - \log_{9}x\). This is because any logarithm base b of b is always equal to 1.
03

Final Simple Form

Since no further simplification is possible and the expression cannot be evaluated without a given value for x, the final expanded form of the given logarithmic expression is \(1 - \log_{9}x\).

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Most popular questions from this chapter

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function.$$ y=\log _{15} x $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{6}\left(\frac{36}{\sqrt{x+1}}\right) $$

Rescarch applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, consumption of natural resources, human memory, progress over time in a sport, profit over time. For the area that you select. explain how logarithmic functions are used and provide examples.

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{7}(7 x) $$
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