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Q41.

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Precalculus Mathematics for Calculus

Found in: Page 352

Book edition 7th Edition

Author(s) James Stewart, Lothar Redlin, Saleem Watson

Pages 948 pages

ISBN 9781337067508

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Short Answer

Logarithmic Equations Use the definition of the logarithmic function to find x.
(a) $\log_{2} (\frac{1}{2}) = x$ (b) $\log_{10} x = - 3$

The required value of x is $- 1$ .
The required value of x is 0.001.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part a. Step 1. Given.

The given equation is $\log_{2} (\frac{1}{2}) = x$ .

Part a. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

Part a. Step 3. Calculation.

We’ll use the definition of the logarithmic function $\log_{a} c = b \Rightarrow a^{b} = c$ .

Comparing $\log_{2} (\frac{1}{2}) = x$ with $\log_{a} c = b$ we get $a = 2, b = x, c = \frac{1}{2}$ .

So, the equivalent exponential form is:

$a^{b} = c$

or, $2^{x} = \frac{1}{2}$

or, $2^{x} = 2^{- 1}$

or, $x = - 1$ [Equated the exponents, since the bases are same]

Hence, the required value of x is $- 1$ .

Part b. Step 1. Given.

The given equation is $\log_{10} x = - 3$ .

Part b. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

Part b. Step 3. Calculation.

We’ll use the definition of the logarithmic function $\log_{a} c = b \Rightarrow a^{b} = c$ .

Comparing $\log_{10} x = - 3$ with $\log_{a} c = b$ we get $a = 10, b = - 3, c = x$ .

So, the equivalent exponential form is:

$a^{b} = c$

or, $10^{- 3} = x$

or, $\frac{1}{1000} = x$

or, $0.001 = x$

Hence, the required value of x is 0.001.

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