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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 57
Chapter 5, Problem 57

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. log3(x+4)=−3

Verified step by step guidance
1
Recall the definition of a logarithm: if \(\log_{a}(b) = c\), then it is equivalent to the exponential form \(a^{c} = b\).
Rewrite the given equation \(\log_{3}(x+4) = -3\) in exponential form: \(3^{-3} = x + 4\).
Calculate \(3^{-3}\) as \(\frac{1}{3^{3}} = \frac{1}{27}\), so the equation becomes \(\frac{1}{27} = x + 4\).
Solve for \(x\) by subtracting 4 from both sides: \(x = \frac{1}{27} - 4\).
Check the domain of the original logarithmic expression: since \(\log_{3}(x+4)\) requires \(x + 4 > 0\), verify that your solution satisfies this condition.

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Key Concepts

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

Properties of Logarithms

Understanding the basic properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable.
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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(f(x)) requires that the argument f(x) be positive. Identifying and applying this domain restriction ensures that any solution found is valid and does not produce undefined expressions.
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Graphs of Logarithmic Functions

Exact and Approximate Solutions

After solving the equation exactly, it is often necessary to find a decimal approximation using a calculator. This helps interpret the solution in a practical context, especially when the exact form is a fraction or irrational number.
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Related Practice
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. 3 ln x - (1/3) ln y

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Textbook Question

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = (1/2)log₂ x

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. 5 ln x - 2 ln y

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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. (1/2)ln x - ln y

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Textbook Question

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log2x

Textbook Question

Graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. f(x) = 2x, g(x) = 2-x