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log_(sqrt5)5=x

$\log _{\sqrt{5}} 5=x$

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Product property of logarithms

Full solution

Q.

\log _{\sqrt{5}} 5=x

Identify Base and Application: Identify the base of the logarithm and what it's being applied to. $\newline$ We have $\log_{\sqrt{5}} 5$ .
Property of Logarithms: Use the property of logarithms that says $\log_b(b) = 1$ . Here, we need to express $5$ as a power of $\sqrt{5}$ .
Express as Power: Recognize that $\sqrt{5}$ is the same as $5^{1/2}$ . So we're looking for a power that makes $5^{1/2}$ equal to $5$ .
Power of a Power Rule: Express $5$ as a power of $5^{(1/2)}$ . $\newline$ $5$ is the same as $(5^{(1/2)})^2$ because $(\sqrt{5})^2 = 5$ .
Use Property Again: Apply the power of a power rule. $\log_{\sqrt{5}}\left((5^{\frac{1}{2}})^2\right)$ simplifies to $2 \cdot \log_{\sqrt{5}}(5^{\frac{1}{2}})$ .
Final Calculation: Use the property from step $2$ . $\newline$ $\log_{\sqrt{5}}(5^{\frac{1}{2}})$ is $1$ because the base and the inside of the log match.
Final Calculation: Use the property from step $2$ . $\log_{\sqrt{5}}(5^{\frac{1}{2}})$ is $1$ because the base and the inside of the log match.Multiply the $1$ by $2$ . So, $x = 2 \times 1$ which is $2$ .

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