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.

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 80

6장, 문제 80

Solve the systems in Exercises 79–80.
${\begin{matrix} log x^{2} = y + 3 \\ log x^{} = y - 1 \end{matrix}$

검증된 단계별 안내

Rewrite the given system of equations for clarity:

\log x^{2} = y + 3

and

\log x = y - 1

Recall the logarithm property:

\log x^{2} = 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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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\).