Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 4 - Section 4.3 - Logarithmic Functions - 4.3 Exercises: 41

Answer

a.) $x = -1$ b.) $x = \frac{1}{1000}$

Work Step by Step

$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$ a.) $\log_2 (\frac{1}{2}) = x$ b.) $\log_{10} x = -3$ a.) $\log_2 (\frac{1}{2}) = x$ Rewrite $\frac{1}{2}$ as $2^{-1}$ [Note:$2^{-1} = \frac{1}{2^1} = \frac{1}{2}$] $$\log_2 2^{-1} = x$$ Use the Third Property of Logarithms: $\log_a a^x = x$ $$\log_2 2^{-1} = x \rightarrow -1 = x$$ $$x=-1$$ b.) $\log_{10} x = -3$ Rewrite the logarithmic form to exponential form: $\log_b a = c \rightarrow b^c = a$ $$\log_{10} x = -3 \rightarrow 10^{-3} = x$$ $$x = \frac{1}{10^3}$$ $$x = \frac{1}{10\times10\times10}$$ $$x = \frac{1}{1000}$$
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