Find Logarithm Base: We need to find the value of log_(9)3. This means we are looking for the exponent that 9 must be raised to in order to get 3.Express 9 in Terms of 3: We know that 9 is 3 squared, i.e., 9 = 3^2. Therefore, we can express 9 in terms of the base 3.Apply Change of Base Formula: Using the change of base formula for logarithms, we can write log_(9)3 as log_(3^2)3.Simplify Using Power Rule: We can simplify log_(3^2)3 by using the power rule of logarithms, which states that log_(b^m)n = (1/m) * log_b(n). Applying this rule, we get log_(3^2)3 = (1/2) * log_3(3).Evaluate log_3(3): We know that log_3(3) is 1, because 3 raised to the power of 1 is 3.Substitute and Simplify: Substituting the value we found in the previous step, we get (1/2) * 1, which simplifies to 1/2.Final Result: Therefore, log_(9)3 = 1/2.

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$\log _{9} 3=$

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Grade 8

Relationship between squares and square roots

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\log _{9} 3=

Find Logarithm Base: We need to find the value of $\log_{9}3$ . This means we are looking for the exponent that $9$ must be raised to in order to get $3$ .
Express $9$ in Terms of $3$ : We know that $9$ is $3$ squared, i.e., $9 = 3^2$ . Therefore, we can express $9$ in terms of the base $3$ .
Apply Change of Base Formula: Using the change of base formula for logarithms, we can write $\log_{9}3$ as $\log_{3^2}3$ .
Simplify Using Power Rule: We can simplify $\log_{3^2}3$ by using the power rule of logarithms, which states that $\log_{b^m}n = \frac{1}{m} \cdot \log_b(n)$ . Applying this rule, we get $\log_{3^2}3 = \frac{1}{2} \cdot \log_3(3)$ .
Evaluate $\log_3(3)$ : We know that $\log_3(3)$ is $1$ , because $3$ raised to the power of $1$ is $3$ .
Substitute and Simplify: Substituting the value we found in the previous step, we get $(\frac{1}{2}) \times 1$ , which simplifies to $\frac{1}{2}$ .
Final Result: Therefore, $\log_{9}3 = \frac{1}{2}$ .