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Chapter 4: Problem 5

Fill in the blank. The \(y\) -axis is a(n) _____________ for the graph of \(f(x)=\log _{a}(x)\).

Short Answer

Expert verified
vertical asymptote

Step by step solution

01

Identify the equation type

The given equation is a logarithmic function in the form of \( f(x) = \log_a(x) \).
02

Recall the properties of the logarithmic function

Recall that for the function \( f(x) = \log_a(x) \), the graph does not intersect the y-axis because the logarithm is undefined for non-positive values of x.
03

Understand the behavior near the y-axis

As \( x \) approaches zero from the positive side, \( f(x) = \log_a(x) \) approaches negative infinity. This indicates that the y-axis acts as a boundary the graph approaches but never touches.
04

Define the asymptote

Based on the behavior near the y-axis and the definition of an asymptote, the y-axis is a vertical asymptote of the logarithmic function \( f(x) = \log_a(x) \).

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Key Concepts

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

Asymptote
When studying logarithmic functions like \(f(x) = \log_a(x)\), it's important to understand what an asymptote is. An asymptote is a line that the graph of a function approaches but never actually touches. In the case of \(f(x) = \log_a(x)\), the y-axis (\(x=0\)) is a vertical asymptote.
Why? As x gets very close to zero from the positive side, \(\log_a(x)\) heads towards negative infinity. This means that the graph will get very close to the y-axis as x decreases, but will never actually reach or intersect the axis. Therefore, the y-axis acts as a boundary that the logarithmic function approaches indefinitely.
Logarithm Properties
To understand logarithmic functions deeply, you'll need to get familiar with some fundamental logarithm properties. Here are a few key properties:
  • Base Relationship: \( \log_a(a) = 1 \) because the log represents the exponent needed to achieve the base value.
  • Product Rule: \( \log_a(xy) = \log_a(x) + \log_a(y) \) breaks down products into sums.
  • Quotient Rule: \( \log_a \left( \frac{x}{y} \right) = \log_a(x) - \log_a(y) \) helps to simplify divisions into differences.
  • Power Rule: \( \log_a(x^k) = k \cdot \log_a(x) \) which pulls power exponents out front.
  • Change of Base Formula: \( \log_a(x) = \frac{\log_b(x)}{\log_b(a)} \) allows conversion between different log bases.
Understanding these properties will help in manipulating and solving equations involving logs.
Undefined Values
One significant property of logarithmic functions, such as \( f(x) = \log_a(x) \), is that they are only defined for positive values of x. This means for any \( x \leq 0 \), \( \log_a(x) \) is undefined.
Why does this happen? Remember, the logarithm \( \log_a(x) \) is the exponent to which the base \( a \) must be raised to produce \( x \). There is no real number solution for this when \( x \) is zero or negative. Since no power of a positive base number can result in zero or a negative number, the log of such values does not exist.
Thus, always ensure x is positive when dealing with logarithmic functions to avoid undefined expressions.

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Most popular questions from this chapter

Solve each problem. To illustrate the "miracle" of compound interest, Ben Franklin bequeathed \(\$ 4000\) to the city of Boston in \(1790 .\) The fund grew to \(\$ 4.5\) million in 200 years. Find the annual rate compounded continuously that would cause this "miracle" to happen.

Visual Magnitude of a Star If all stars were at the same distance, it would be a simple matter to compare their brightness. However, the brightness that we see, the apparent visual magnitude \(m,\) depends on a star's intrinsic brightness, or absolute visual magnitude \(M_{V},\) and the distance \(d\) from the observer in parsecs ( 1 parsec \(=3.262\) light years), according to the formula \(m=M_{V}-5+5 \cdot \log (d) .\) The values of \(M_{V}\) range from \(-8\) for the intrinsically brightest stars to \(+15\) for the intrinsically faintest stars. The nearest star to the sun, Alpha Centauri, has an apparent visual magnitude of 0 and an absolute visual magnitude of 4.39 . Find the distance \(d\) in parsecs to Alpha Centauri.

Solve each problem. Find the annual percentage rate compounded continuously to the nearest tenth of a percent for which \(\$ 10\) would grow to \(\$ 30\) for each of the following time periods. a. 5 years b. 10 years c. 20 years d. 40 years

Solve \(\log _{2}(x-3)=8.\)

Solve each problem. When needed, use 365 days per year and 30 days per month. Population Growth The function \(P=2.4 e^{0.03 t}\) models the size of the population of a small country, where \(P\) is in millions of people in the year \(2000+t\) a. What was the population in \(2000 ?\) b. Use the formula to estimate the population to the nearest tenth of a million in 2020 .
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