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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. 193 (page 1034)

In the following exercises,solve each logarithmic equation.
$\log_{a} 24 = 3$

Short Answer

Expert verified

The solution $a = \sqrt[3]{24}$ .

Step by step solution

01

Step 1.Given information

Consider the logarithmic equation $\log_{a} 24 = 3$ .

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

02

Step 2.Find the exponential form

Comparing $\log_{a} 24 = 3$ with the standard form $y = \log_{a} x$ ,you have

$y = 3, a = a, x = 24$

Convert logarithmic equation in exponential equation as shown below

$x = a^{y} 24 = a^{3} a^{3} = 24 S o, t h e e q u a t i o n i n e x p o n e n t i a l f o r m i s a^{3} = 24$

03

Step 3.Solve the equation and find the value of a.

${(a^{3})}^{\frac{1}{3}} = {(24)}^{\frac{1}{3}} [R a i s e r a d i c a l \frac{1}{3} t o t h e b o t h s i d e s] a = \sqrt[3]{24} S o, t h e s o l u t i o n i s a = \sqrt[3]{24}$

04

Step4.Check

To check the solution,replace a by $\sqrt[3]{24}$ in original equation

$\log_{a} 24 = 3 \log_{\sqrt[3]{24}} 24 = 3 [a = \sqrt[3]{24}] {(\sqrt[3]{24})}^{3} = 24 24 = 24 [t r u e] T h e r e f o r e, t h e s o l u t i o n i s a = \sqrt[3]{24}$

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