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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. 10.41 (page 1024)

Find the exact value of each logarithm without using a calculator:
$a) \log_{12} (144) b) \log_{4} (2) c) \log_{2} (\frac{1}{32})$

Short Answer

Expert verified

Part (a) The value of the logarithm is $2$ .

Part (b) The value of the logarithm is $\frac{1}{2}$ .

Part (c) The value of the logarithm is $- 5$ .

Step by step solution

01

Step 1. Given information.

The given logarithms are

$a) \log_{12} (144) b) \log_{4} (2) c) \log_{2} (\frac{1}{32})$

02

Step 2. Solving part (a).

Consider the given expression,

$\log_{12} (144) = x$

Now, we will convert the given expression to exponential form and simplify.

$\log_{12} (144) = x {(12)}^{x} = 144 {(12)}^{x} = {(12)}^{2} x = 2$

Hence, the value of the logarithm is $2$ .

03

Step 3. Solving part (b).

Consider the given expression,

$\log_{4} (2) = x$

Now, we will convert the given expression to exponential form and simplify.

$\log_{4} (2) = x 4^{x} = 2 {(2^{2})}^{x} = 2 2^{2 x} = 2^{1} 2 x = 1 x = \frac{1}{2}$

Hence, the value of the logarithm is $\frac{1}{2}$ .

04

Step 4. Solving part (c).

Consider the given expression,

$\log_{2} (\frac{1}{32}) = x$

Now, we will convert the given expression to exponential form and simplify.

$\log_{2} (\frac{1}{32}) = x 2^{x} = \frac{1}{32} 2^{x} = \frac{1}{2^{5}} 2^{x} = 2^{- 5} x = - 5$

Hence, the value of the logarithm is $- 5$ .

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