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Chapter 3: Problem 28

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{9}$$

Short Answer

Expert verified
The value of \(\log _{3} \frac{1}{9}\) is -2.

Step by step solution

01

Convert to exponential form

We start by expressing the given logarithm, \(\log _{3} \frac{1}{9}\), in its equivalent exponential form. This means the base of the logarithm (\(3\)) raised to some power we need to solve for (\(\log _{3} \frac{1}{9}\)) equals \(\frac{1}{9}\). Therefore, our exponential equation will be \(3^y = \frac{1}{9}\).
02

Simplify the equation

The second step is to simplify the equation. Notice that \(\frac{1}{9}\) can be expressed as \(3^{-2}\) because any number raised to the power of -2 is equal to its reciprocal squared. Then we have \(3^y = 3^{-2}\). Since the bases on both sides of the equation are the same (\(3\)), it can be said that the exponents must also be the same, which gives \(y = -2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Form
When dealing with logarithms, converting them to exponential form can simplify the problem. In the exercise, we start with the logarithmic expression \( \log_{3} \frac{1}{9} \). To convert this, remember that a logarithm answers the question: "To what exponent must the base be raised to produce a given number?"

Here's a simple breakdown:
  • The base \(3\) raised to the unknown exponent equals \( \frac{1}{9} \).
  • This gives us the equation \( 3^y = \frac{1}{9} \).
So, translating a log expression into an exponential equation helps visualize the relationship between the base and the exponent needed to produce the given number. Mastering this step allows us to handle more complex logarithmic problems with ease.
Logarithmic Equations
Logarithmic equations involve the unknown being part of a logarithm. To solve, we often convert the logarithm into an exponential form. This exercise shows this technique precisely.

Why convert? Because exponential equations are typically simpler to solve. Once in the form \( 3^y = 3^{-2} \), we can see a clear path to finding \( y \).
  • Equating the exponents when the bases are the same simplifies the problem.
  • It turns the logarithmic problem into basic algebra.
Understanding logarithmic equations is beneficial as it provides a tool to decode exponential relationships present in logarithms. Such understanding can powerfully simplify both academic problems and real-world scenarios involving exponential growth or decay.
Exponents
Exponents are a fundamental part of mathematics, expressing repeated multiplication. In our exercise, recognizing \( \frac{1}{9} \) as \( 3^{-2} \) involved understanding the concept of negative exponents.

Here's what you need to know:
  • Positive exponents indicate multiplication, like \( 3^2 = 9 \).
  • Negative exponents indicate the reciprocal, so \( a^{-b} = \frac{1}{a^b} \).
Using these principles, we set up the equation correctly: \( 3^y = 3^{-2} \).

From here, since the bases are the same, it's straightforward that \( y = -2 \).

Grasping how exponents work allows a deeper understanding of many mathematical principles, from simple calculations to advanced scientific computations. It's a crucial skill that applies to various domains beyond math.

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Most popular questions from this chapter

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because carbon-14 decays exponentially, carbon dating can determine the ages of ancient fossils.

Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.

In Exercises \(125-132,\) 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.. $$3^{x}=2 x+3$$
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