What is the value of log_(3)root(3)(9) ? Answer:

Understand the Problem: Understand the problem. We need to find the value of the logarithm of the cube root of 9 to the base 3.Express as Power of 9: Express the cube root of 9 as a power of 9. The cube root of 9 can be written as 9^(1/3).Rewrite as Power of 3: Rewrite 9 as a power of 3. Since 9 is 3 squared (3^2), we can write 9^(1/3) as (3^2)^(1/3).Apply Power Rule: Apply the power rule for exponents (a^(m))^n = a^(m*n). (3^2)^(1/3) becomes 3^(2*(1/3)), which simplifies to 3^(2/3).Evaluate Logarithm: Evaluate the logarithm. Now we have log_3(3^(2/3)). Since the base of the logarithm and the base of the exponent are the same, the logarithm of a power is the exponent. Therefore, log_3(3^(2/3)) = 2/3.

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What is the value of
log_(3)root(3)(9) ?
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What is the value of $\log _{3} \sqrt[3]{9}$ ? $\newline$ Answer:

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Q. What is the value of

\log _{3} \sqrt[3]{9}

\newline

Answer:

Understand the Problem: Understand the problem. $\newline$ We need to find the value of the logarithm of the cube root of $9$ to the base $3$ .
Express as Power of $9$ : Express the cube root of $9$ as a power of $9$ . $\newline$ The cube root of $9$ can be written as $9^{1/3}$ .
Rewrite as Power of $3$ : Rewrite $9$ as a power of $3$ . $\newline$ Since $9$ is $3$ squared ( $3^2$ ), we can write $9^{1/3}$ as $(3^2)^{1/3}$ .
Apply Power Rule: Apply the power rule for exponents $(a^{m})^{n} = a^{m*n}$ . $(3^{2})^{\frac{1}{3}}$ becomes $3^{2*\frac{1}{3}}$ , which simplifies to $3^{\frac{2}{3}}$ .
Evaluate Logarithm: Evaluate the logarithm. $\newline$ Now we have $\log_3(3^{2/3})$ . Since the base of the logarithm and the base of the exponent are the same, the logarithm of a power is the exponent. $\newline$ Therefore, $\log_3(3^{2/3}) = \frac{2}{3}$ .