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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q 32. (page 298)

Find the exact value of the logarithm without using a calculator.
$\log_{5} (\sqrt[3]{25})$

Short Answer

Expert verified

The exact value of the logarithm is $\frac{2}{3}$ .

Step by step solution

01

Step 1. Given information

A logarithmic function is given as $\log_{5} (\sqrt[3]{25})$ .

02

Step 2. Formula Used

The logarithmic function $y = \log_{a} (x)$ can be changed into an exponential statement as $x = a^{y}$ .

03

Step 3. Finding Value of the logarithm

Consider $\log_{5} (\sqrt[3]{25}) = x$

By changing rule of logarithm to exponential function,

$5^{x} = \sqrt[3]{25} 5^{x} = {(5^{2})}^{\frac{1}{3}} 5^{x} = 5^{\frac{2}{3}} x = \frac{2}{3} \log_{5} (\sqrt[3]{25}) = \frac{2}{3}$

The required value is,

$\log_{5} (\sqrt[3]{25}) = \frac{2}{3}$

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