Problem 41 - Essential Calculus Early Transcendentals

Question

(a) How is the logarithmic function \(y=\log _{a} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y \(=\log _{a} x\) if \(a>1\)

Step-by-Step Solution

Answer

(a) \(y=\log_a x\) is the inverse of \(a^y = x\). (b) Domain: \(x > 0\). (c) Range: all real numbers. (d) The graph is an increasing curve for \(a > 1\).

Step 1: Understanding the Definition of a Logarithmic Function

The logarithmic function \(y=\log_{a}x\) is defined as the inverse function of the exponential function \(a^y = x\). This means that for every real number \(x>0\), \(y\) is the power to which \(a\) must be raised to obtain \(x\). Hence, \(y=\log_{a}x\) can be rewritten as \(a^y = x\).

Step 2: Determining the Domain of the Logarithmic Function

The domain of the logarithmic function \(y = \log_{a}x\) is defined by the set of all positive real numbers 'x'. This is because the logarithm of zero or a negative number is undefined, so we require \(x > 0\) for the function to exist.

Step 3: Identifying the Range of the Logarithmic Function

The range of the logarithmic function \(y = \log_{a}x\) is all real numbers. Because as \(x\) approaches zero from the positive side, \(y\) (the output of the function) approaches negative infinity, and as \(x\) increases towards infinity, \(y\) also increases towards infinity.

Step 4: Sketching the Graph of the Logarithmic Function for \(a>1\)

The graph of \(y = \log_{a}x\) for \(a>1\) is an increasing curve that passes through the point (1,0) on the Cartesian plane. The curve is asymptotic to the y-axis, which means it gets closer to the y-axis as \(x\) approaches zero, but it never crosses or touches it. As \(x\) increases, the curve continues to rise slowly and steadily. This behavior reflects that, for \(a>1\), the function is always increasing.

Key Concepts

Inverse Functions

An inverse function essentially swaps inputs and outputs, reversing the effect of the original function. For logarithmic functions, we start with the exponential function, where we express a number as an exponent of a base. Consider the exponential function \( a^y = x \). Here, if we think of \( a \) as a number greater than zero and not equal to one, the value you get by raising \( a \) to some power \( y \) equals \( x \).

Inverting this relationship, we define a logarithm: \( y = \log_a x \). With this, \( y \) is the exponent to which the base \( a \) must be raised to produce \( x \). This makes the logarithmic function the inverse of the exponential function. In simpler terms:

• If \( a^y = x \), then \( y = \log_a x \).
• They "undo" each other; if you log an exponentiated value, you return to the original exponent.

This explains why understanding exponential behavior helps in grasping logarithmic functions.

Domain and Range

The domain and range of a logarithmic function are fundamental in understanding where the function exists and what values it can take.

• Domain: The domain of \( y = \log_a x \) consists of all positive real numbers. Simply put, \( x \) must be greater than 0, as you cannot take the logarithm of zero or a negative number. Mathematically, this is expressed as \( x > 0 \).
• Range: The range of the logarithmic function is all real numbers. As \( x \) approaches zero from the positive side, \( y \) approaches negative infinity. Conversely, as \( x \) continues to increase beyond any bound, \( y \) will tend towards positive infinity.

This behavior highlights how logarithmic functions manage to maintain a continuous output across infinitely varied input values.

Graph of Logarithmic Functions

Understanding the graph of a logarithmic function provides visual intuition of its behavior. For \( a > 1 \), the graph of \( y = \log_a x \) presents distinct traits:

• The curve is steadily increasing. It smoothly rises from the lower left part of the graph towards the upper right.
• The graph passes through the point (1, 0). This is because \( \log_a 1 = 0 \) for any positive \( a \).
• The y-axis (\( x = 0 \)) acts as a vertical asymptote. As \( x \) gets closer to zero, the graph moves infinitely close to the vertical line without actually meeting it, showing that logarithmic values get very large negative outputs as their inputs become very small.

These characteristics reflect the continuously increasing nature of the function with heavier inclination at smaller \( x \) values, gradually leveling out as \( x \) grows.