Question
valuate using the properties of logarithms: \( 10^{\log \sqrt{5}} \)
Ask by Collins Huff. in the United States
Nov 02,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 10^{\log \sqrt{5}} = \sqrt{5} \)
Solution
Calculate or simplify the expression \( 10^{\log \sqrt{5}} \).
Calculate the value by following steps:
- step0: Calculate:
\(10^{\log_{10}{\left(\sqrt{5}\right)}}\)
- step1: Simplify:
\(10^{\frac{1}{2}\log_{10}{\left(5\right)}}\)
- step2: Transform the expression:
\(10^{\log_{10}{\left(5^{\frac{1}{2}}\right)}}\)
- step3: Transform the expression:
\(5^{\frac{1}{2}}\)
- step4: Transform the expression:
\(\sqrt{5}\)
The value of \(10^{\log \sqrt{5}}\) is \(\sqrt{5}\).
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Simplify this solution Beyond the Answer
Let’s break it down! The expression \( 10^{\log \sqrt{5}} \) utilizes the power of exponents and logarithms. First, we can simplify \( \log \sqrt{5} \) using the logarithmic property \( \log a^{b} = b \log a \). This gives us: \[ \log \sqrt{5} = \log 5^{1/2} = \frac{1}{2} \log 5. \] Now, substituting that into the original expression, we have: \[ 10^{\log \sqrt{5}} = 10^{\frac{1}{2} \log 5}. \] Using the property of exponents, we can rewrite this as: \[ = (10^{\log 5})^{1/2} = 5^{1/2} = \sqrt{5}. \] So, you find that \( 10^{\log \sqrt{5}} = \sqrt{5} \). Clever, right?
