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Chapter 12: Problem 9

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log \left(\frac{x}{100}\right)$$

Short Answer

Expert verified
\(\log (x) - 2\)

Step by step solution

01

Identify the Property

We identify the property of logarithms that will be used. The expression inside the log is a fraction, showing a division operation. The apt property to apply in such a scenario is 'log(a/b) = log(a) - log(b)'.
02

Apply the Property

Let's apply the property to the given expression \( \log \left(\frac{x}{100}\right) \). This expands to \( \log (x) - \log (100) \).
03

Simplify the Expression

The logarithm base 10 of 100 is 2, as 10^2 = 100. Hence, we can simplify \( \log (100) \) as 2. So, the final expanded expression is \( \log (x) - 2 \).

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Most popular questions from this chapter

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?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Simplify: \(\left(-2 x^{3} y^{-2}\right)^{-4}\)

Describe the quotient rule for logarithms and give an example.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(3 x-2)=2$$
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