Question
Use properties of logarithms to evaluate the expression. \( 2 \log 5+\log 4 \) \( \square \)
Ask by Bartlett Nguyen. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression simplifies to 2.
Solution
Calculate or simplify the expression \( 2 \log(5)+\log(4) \).
Calculate the value by following steps:
- step0: Calculate:
\(2\log_{10}{\left(5\right)}+\log_{10}{\left(4\right)}\)
- step1: Rewrite the expression:
\(\log_{10}{\left(5^{2}\right)}+\log_{10}{\left(4\right)}\)
- step2: Use the logarithm product rule:
\(\log_{10}{\left(5^{2}\times 4\right)}\)
- step3: Multiply the numbers:
\(\log_{10}{\left(100\right)}\)
- step4: Write in exponential form:
\(\log_{10}{\left(10^{2}\right)}\)
- step5: Simplify the expression:
\(2\)
The expression \(2 \log 5 + \log 4\) simplifies to \(2\).
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Simplify this solution Beyond the Answer
To evaluate the expression \( 2 \log 5 + \log 4 \), we can utilize the properties of logarithms. First, we can rewrite \( 2 \log 5 \) as \( \log(5^2) \), which simplifies to \( \log 25 \). Therefore, the expression becomes: \[ \log 25 + \log 4 \] By applying the property \( \log a + \log b = \log(ab) \), we can combine these logarithms: \[ \log(25 \times 4) = \log 100 \] Now, since \( \log 100 = 2 \) (because \( 100 = 10^2 \)), we conclude: \[ 2 \log 5 + \log 4 = 2 \]
