Find logarithm base and exponent: We need to find the value of log_(27)9. This means we are looking for the exponent that 27 must be raised to in order to get 9.Recognize powers of 3: Recognize that 27 is a power of 3, specifically 27 = 3^3, and 9 is also a power of 3, specifically 9 = 3^2.Rewrite logarithm in base 3: Rewrite the logarithm in terms of the base 3: log_(3^3)(3^2).Apply logarithm property: Use the property of logarithms that says log_(a^c)(b^d) = d/c * log_a(b). In this case, log_(3^3)(3^2) = 2/3 * log_3(3).Evaluate logarithm: Since log_3(3) is equal to 1 (because 3 raised to the power of 1 is 3), we have 2/3 * 1 = 2/3.Final result: Therefore, log_(27)9 = 2/3.

Grade 8

Relationship between squares and square roots

Full solution

\log _{27} 9=

Find logarithm base and exponent: We need to find the value of $\log_{27}9$ . This means we are looking for the exponent that $27$ must be raised to in order to get $9$ .
Recognize powers of $3$ : Recognize that $27$ is a power of $3$ , specifically $27 = 3^3$ , and $9$ is also a power of $3$ , specifically $9 = 3^2$ .
Rewrite logarithm in base $3$ : Rewrite the logarithm in terms of the base $3$ : $\log_{3^3}(3^2)$ .
Apply logarithm property: Use the property of logarithms that says $\log_{a^c}(b^d) = \frac{d}{c} \cdot \log_a(b)$ . In this case, $\log_{3^3}(3^2) = \frac{2}{3} \cdot \log_3(3)$ .
Evaluate logarithm: Since $\log_3(3)$ is equal to $1$ (because $3$ raised to the power of $1$ is $3$ ), we have $\frac{2}{3} \times 1 = \frac{2}{3}$ .
Final result: Therefore, $\log_{27}9 = \frac{2}{3}$ .