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Chapter 26: Chapter 26

Problem 754

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Express the logarithm of 7 to the base 3 in terms of common logarithms.

Short Answer

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The logarithm of 7 to the base 3 in terms of common logarithms is \(\log_3 7 = \frac{\log 7}{\log 3}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Identify the given logarithm and the desired base

We are given the logarithm \(\log_3 7\) and we want to express it in terms of common logarithms, which are base 10.

Step 2: Apply the change of base formula

Now we will use the change of base formula to convert our given logarithm to base 10. \(\log_{a}b = \frac{\log_{c}b}{\log_{c}a}\) In our case, \(a = 3\), \(b = 7\), and \(c = 10\). So, we have: \(\log_3 7 = \frac{\log_{10} 7}{\log_{10} 3}\)

Step 3: Simplify the expression

We can now use the simplified notation for common logarithms, which is \(\log b = \log_{10} b\). So our expression becomes: \(\log_3 7 = \frac{\log 7}{\log 3}\)

Step 4: Write the final answer

Now we have expressed the logarithm of 7 to the base 3 in terms of common logarithms: \(\log_3 7 = \frac{\log 7}{\log 3}\)

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