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Ch. 5 - Systems of Equations and Inequalities
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. 5 - Systems of Equations and InequalitiesProblema 80
Capitolo 6, Problema 80

Solve the systems in Exercises 79–80.
{log x2=y+3log x=y−1\(\left\)\{\(\begin{array}{l}\[\text{log }\)x^2=y+3\\ \(\text{log }\)x^{}=y-1\(\end{array}\]\right\).

Guida verificata passo dopo passo
1
Rewrite the given system of equations for clarity: logx2 = y + 3 and logx = y - 1.
Recall the logarithm property: logx2 = 2 log x. Use this to rewrite the first equation as 2 log x = y + 3.
Substitute log x from the second equation into the first. Since log x = y - 1, replace log x in the first equation with y - 1 to get 2(y - 1) = y + 3.
Solve the resulting linear equation for y: expand and simplify 2y - 2 = y + 3, then isolate y.
Once you find y, substitute it back into log x = y - 1 to find log x, and then solve for x by rewriting the logarithmic equation in exponential form.

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Properties of Logarithms

Understanding the properties of logarithms, such as the power rule (log a^b = b log a), is essential for manipulating and simplifying logarithmic expressions. This allows rewriting terms like log x^2 as 2 log x, facilitating easier comparison and solving of equations.
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Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy all given equations simultaneously. Techniques include substitution and elimination, which help reduce the system to a single-variable equation for easier solution.
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Solving Systems of Equations - Substitution

Relationship Between Logarithmic and Linear Expressions

Recognizing how logarithmic expressions relate to linear equations is crucial. For example, expressing log x in terms of y allows converting the system into linear form, making it easier to solve using algebraic methods.
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Logarithms Introduction
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