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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 19
Capitolo 5, Problema 19

Write each equation in its equivalent logarithmic form. 7y = 200

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Identify the given exponential equation: \(7^y = 200\).
Recall the definition of logarithms: if \(a^x = b\), then the equivalent logarithmic form is \(\log_a b = x\).
In this problem, the base \(a\) is 7, the exponent \(x\) is \(y\), and the result \(b\) is 200.
Rewrite the equation \(7^y = 200\) in logarithmic form using the definition: \(\log_7 200 = y\).
This expresses the original exponential equation as a logarithmic equation, which is the equivalent form requested.

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Exponential and Logarithmic Forms

Exponential and logarithmic forms are two ways to express the same relationship. An equation like a^y = b can be rewritten in logarithmic form as log_a(b) = y, where the base a remains the same. Understanding this equivalence is essential for converting between forms.
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Solving Logarithmic Equations

Definition of a Logarithm

A logarithm answers the question: to what power must the base be raised to produce a given number? For example, log_a(b) = y means a raised to y equals b. This definition is fundamental for rewriting exponential equations as logarithms.
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Logarithms Introduction

Properties of Logarithms

Logarithms have specific properties, such as the base must be positive and not equal to 1, and the argument must be positive. These properties ensure the logarithmic expression is valid and help in correctly rewriting and solving equations.
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Change of Base Property
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The graph of an exponential function is given. Select the function for each graph from the following options:

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In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \\ f(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

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