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Chapter 13: Problem 21

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{5} 25 t$$

Short Answer

Expert verified
The short answer is: \(\log_{5} 25t = 2 + \log_{5} t\).

Step by step solution

01

Apply the Logarithm Product Rule

Recall that the logarithm product rule states that \(\log_{b} (xy) = \log_{b}x + \log_{b}y\). Using this rule, rewrite the given logarithm as a sum of simpler logarithms: \(\log_{5} 25t = \log_{5} 25 + \log_{5} t\)
02

Simplify the Logarithm of the Constant Term

Considering the first term, \(\log_{5} 25\), we know that \(5^2 = 25\). So, the base 5 logarithm of 25 is 2. Substitute the simplified value for the first term: \(\log_{5} 25 + \log_{5} t = 2 + \log_{5} t\)
03

Write the Final Simplified Expression

The simplified expression, expressed as the sum of simpler logarithms, is: \(2 + \log_{5} t\)

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