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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 191 (page 1034)

In the following exercises,solve each logarithmic equation.
$\log_{a} 27 = 3$ .

Short Answer

Expert verified

a=3

Step by step solution

01

Step 1.Given information

Consider the logarithmic equation $\log_{a} 27 = 3$

The logarithmic from $\log_{a} 27 = 3$ is equivalent to exponential form $x = a^{y}$ .

02

Step 2.Solve the equation and find the value of a.

Comparing $\log_{a} 27 = 3$ with the standard form $y = \log_{a} x$ ,you have

Convert logarithmic equation in exponential equation as shown below

$x = a^{y} 27 = a^{3} a^{3} = 27 S o, t h e e q u a t i o n i n e x p o n e n t i a l f o r m i s a^{3} = 27 .$

Now solve the equation as

role="math" localid="1644925625777" ${(a^{3})}^{\frac{1}{3}} = {(27)}^{\frac{1}{3}} [R a i s e R a d i c a l] {(a^{3})}^{\frac{1}{3}} = {(3^{3})}^{\frac{1}{3}} {(a)}^{3 \frac{1}{3}} = {(3)}^{3 \frac{1}{3}} a = 3 S o, t h e s o l u t i o n i s a = 3$

To check the solution,replace a by 3 in original equation

$\log_{a} 27 = 3 \log_{3} 27 = 3 [a = 3] 3^{3} = 27 27 = 27 [t r u e] T h e r e f o r e, t h e s o l u t i o n i s a = 3$

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Most popular questions from this chapter

In the following exercises, solve each logarithmic equation.

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