Solve the systems in Exercises 79–80.{log⁡y x=3log⁡y (4x)=5\left\{\begin{array}{l}\log_{y}\text{ x=3}\\ \log_{y}\text{ (4x)=5}\end{array}\right.

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 79

6장, 문제 79

Solve the systems in Exercises 79–80.
${\begin{matrix} \log_{y} x=3 \\ \log_{y} (4x)=5 \end{matrix}$

검증된 단계별 안내

Start by writing down the given system of logarithmic equations:

\log y x = 3

and

\log y (4x) = 5

Recall the definition of logarithms:

\log a b = c

means

a^c = b

. Use this to rewrite each logarithmic equation in exponential form.

Rewrite the first equation:

y^3 = x

. Rewrite the second equation:

y^5 = 4x

Substitute the expression for

x

from the first equation into the second equation: replace

x

y^5 = 4x

with

y^3

, resulting in

y^5 = 4y^3

Solve the resulting equation for

y

by dividing both sides by

y^3

(assuming

y 
eq 0

), which gives

y^{5-3} = 4

y^2 = 4

. Then find the possible values of

y

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주요 개념

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Logarithmic Functions and Their Properties

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding properties like log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x) is essential for manipulating and solving logarithmic equations.

Change of Base and Variable Identification

In equations involving logs with unknown bases or arguments, recognizing how to express variables and rewrite equations using properties or substitutions helps isolate variables. Here, identifying y as the base and expressing x in terms of y is key to solving the system.

Solving Systems of Equations

A system of equations involves finding values that satisfy all equations simultaneously. Techniques include substitution or elimination. For logarithmic systems, converting logs to exponential form often simplifies the process and reveals relationships between variables.

Solve the systems in Exercises 79–80.{logy x=3logy (4x)=5\(\left\)\{\(\begin{array}{l}\]\log\)_{y}\(\text{ x=3}\)\\ \(\log\)_{y}\(\text{ (4x)=5}\[\end{array}\]\right\).