Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 63

Chapter 5, Problem 63

Give the equation of each exponential function whose graph is shown.

Verified step by step guidance

Step 1: Identify the general form of the exponential function. Since the graph shows an exponential function with negative values, it likely has the form $y = -a e^{bx}$, where $a > 0$ and $b$ is a constant to be determined.

Step 2: Use the point where the graph crosses the y-axis, which is $(0, -4)$. Substitute $x=0$ and $y=-4$ into the equation $y = -a e^{bx}$ to find $a$. Since $e^{0} = 1$, this gives $-4 = -a \cdot 1$, so $a = 4$.

Step 3: Now the equation is $y = -4 e^{bx}$. Use another point from the graph, for example $(-4, -\frac{4}{e})$, and substitute $x = -4$ and $y = -\frac{4}{e}$ into the equation to solve for $b$.

Step 4: Substitute the values into the equation: $-\frac{4}{e} = -4 e^{b(-4)}$. Simplify this to $\frac{4}{e} = 4 e^{-4b}$, then divide both sides by 4 to get $\frac{1}{e} = e^{-4b}$.

Step 5: Recognize that $\frac{1}{e} = e^{-1}$, so set $e^{-1} = e^{-4b}$. Since the bases are the same, equate the exponents: $-1 = -4b$, and solve for $b$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Function Form

An exponential function is generally written as y = ab^x, where a is the initial value (y-intercept) and b is the base that determines the growth or decay rate. Understanding this form helps in identifying the function from given points on its graph.

Using Points to Find Parameters

Given points on the graph, you can substitute their coordinates into the exponential function to create equations. Solving these equations allows you to find the values of a and b, which define the specific exponential function.

Properties of the Number e

The number e (~2.718) is a special base for exponential functions, often used in continuous growth or decay models. Recognizing e in the points (like -4/e or -4e) helps in simplifying and understanding the function's behavior.

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The Number e

Related Practice

Textbook Question

In Exercises 60–63, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. (log2 x)^4 = 4 log2 x

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Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. $$\frac{1}{2}$ $\left$( $\log$_5 x + $\log$_5 y $\right$) - 2 $\log$_5 (x + 1)$

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Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

g(x) = 1-log x

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Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 6+2 ln x=5