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Chapter 4: Problem 58

What is an exponential function?

Short Answer

Expert verified
An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \) where 'a' and 'b' are constants, 'b' is greater than zero and not equal to one, and 'x' is a variable.

Step by step solution

01

Define Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \), where 'a' and 'b' are real numbers, and \( b > 0, b \neq 1 \), and 'x' is any real number. The base, 'b', is constant and the exponent, 'x', is a variable.
02

Explain Characteristics of Exponential Function

Exponential functions are characterized by their rapid growth or decay, depending on the value of the base 'b'. If \( b > 1 \), the function 'f(x)' represents exponential growth, while if \( 0 < b < 1 \), it represents exponential decay.
03

Provide an Example of Exponential Function

A simple example of an exponential function is \( f(x) = 2^x \). Here, the base is '2', which is a constant and the exponent 'x' is the variable. As 'x' increases, the function f(x) grows exponentially.

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Most popular questions from this chapter

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. $$ \log _{5}\left(\frac{125}{y}\right) $$

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln (x+6)-3 \ln x $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 5+\log 2 $$

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. $$ \log M^{-8} $$
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