Domanda del libro di testo
Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3x and g(x) = 3-x
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 49
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. log3x=4
Guida verificata passo dopo passo1
Identify the given logarithmic equation: \(\log_{3} x = 4\).
Recall the definition of a logarithm: \(\log_{a} b = c\) means \(a^{c} = b\). Using this, rewrite the equation as \(3^{4} = x\).
Calculate the value of \$3^{4}$ to find $x$. (You can leave it as an expression for the exact answer.)
Check the domain of the original logarithmic expression. Since \(\log_{3} x\) is defined only for \(x > 0\), ensure your solution satisfies this condition.
If needed, use a calculator to find the decimal approximation of \(x\) to two decimal places.
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Definition of Logarithms
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₃(x) = 4 means 3 raised to the power 4 equals x. Understanding this definition allows you to rewrite logarithmic equations in exponential form to solve for the variable.
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Logarithms Introduction
Domain of Logarithmic Functions
The domain of a logarithmic function log_b(x) includes only positive real numbers (x > 0) because logarithms of zero or negative numbers are undefined. When solving logarithmic equations, it is essential to check that solutions fall within this domain to ensure they are valid.
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5:26Graphs of Logarithmic Functions
Exact and Approximate Solutions
Logarithmic equations often yield exact solutions expressed in exponential form. However, when a decimal approximation is required, a calculator can be used to find a numerical value, typically rounded to a specified number of decimal places, such as two decimals for clarity and precision.
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4:20Graph Hyperbolas at the Origin
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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. log x + 3 log y
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