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Chapter 8: Problem 2

Write the following using logarithms: (a) \(10^{2}=100\) (b) \(0.001=10^{-3}\) (c) \(\mathrm{e}^{-1.3}=0.2725\) (d) \(\mathrm{e}^{1.5}=4.4817\)

Short Answer

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Question: Rewrite the following equations using logarithms: (a) \(10^{2}=100\) (b) \(0.001=10^{-3}\) (c) \(\mathrm{e}^{-1.3}=0.2725\) (d) \(\mathrm{e}^{1.5}=4.4817\) Answer: (a) \(\log_{10}(100)=2\) (b) \(\log_{10}(0.001)=-3\) (c) \(\ln(0.2725)=-1.3\) (d) \(\ln(4.4817)=1.5\)

Step by step solution

01

(a) Rewrite the equation using logarithms

To write the equation \(10^{2}=100\) using logarithms, we simply take the logarithm of both sides, with base 10. This gives us: \(\log_{10}(10^2) = \log_{10}(100)\). Using the property of logarithms that states \(\log_{b}(b^x) = x\), we can simplify the left side of the equation: \(2=\log_{10}(100)\) So, the equation \(10^{2}=100\) can be written using logarithms as \(\log_{10}(100)=2\).
02

(b) Rewrite the equation using logarithms

To rewrite the equation \(0.001=10^{-3}\) using logarithms, we take the logarithm of both sides with base 10. This gives us: \(\log_{10}(0.001) = \log_{10}(10^{-3})\). Using the property of logarithms that states \(\log_{b}(b^x) = x\), we can simplify the right side of the equation: \(\log_{10}(0.001) = -3\) So, the equation \(0.001=10^{-3}\) can be written using logarithms as \(\log_{10}(0.001)=-3\).
03

(c) Rewrite the equation using logarithms

To rewrite the equation \(\mathrm{e}^{-1.3}=0.2725\) using logarithms, we take the natural logarithm (logarithm with base \(\mathrm{e}\)) of both sides. This gives us: \(\ln(\mathrm{e}^{-1.3}) = \ln(0.2725)\). Using the property of logarithms that states \(\ln(\mathrm{e}^x) = x\), we can simplify the left side of the equation: \(-1.3 = \ln(0.2725)\) So, the equation \(\mathrm{e}^{-1.3}=0.2725\) can be written using logarithms as \(\ln(0.2725)=-1.3\).
04

(d) Rewrite the equation using logarithms

To rewrite the equation \(\mathrm{e}^{1.5}=4.4817\) using logarithms, we take the natural logarithm (logarithm with base \(\mathrm{e}\)) of both sides. This gives us: \(\ln(\mathrm{e}^{1.5}) = \ln(4.4817)\). Using the property of logarithms that states \(\ln(\mathrm{e}^x) = x\), we can simplify the left side of the equation: \(1.5 = \ln(4.4817)\) So, the equation \(\mathrm{e}^{1.5}=4.4817\) can be written using logarithms as \(\ln(4.4817)=1.5\).

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