Solve for the exact value of x. log_(2)(2x)+3log_(2)(3)=0 Answer:

Apply power rule: Apply the power rule of logarithms to the term 3log₂(3). The power rule states that a*log_b(c) = log_b(c^a), so we can rewrite 3log₂(3) as log₂(3^3). log₂(2x) + log₂(3^3) = 0Combine logarithmic terms: Combine the logarithmic terms using the product rule. The product rule states that log_b(m) + log_b(n) = log_b(m*n), so we can combine the terms. log₂(2x * 3^3) = 0Simplify expression: Simplify the expression inside the logarithm. 2x * 3^3 = 2x * 27 log₂(54x) = 0Convert to exponential: Convert the logarithmic equation to an exponential equation. If log₂(54x) = 0, then 2^0 = 54x. 1 = 54xSolve for x: Solve for x. Divide both sides by 54 to isolate x. x = 1/54

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Find derivatives of logarithmic functions

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Q. Solve for the exact value of

x

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\log _{2}(2 x)+3 \log _{2}(3)=0

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Answer:

Apply power rule: Apply the power rule of logarithms to the term $3\log_2(3)$ . The power rule states that $a\log_b(c) = \log_b(c^a)$ , so we can rewrite $3\log_2(3)$ as $\log_2(3^3)$ . $\log_2(2x) + \log_2(3^3) = 0$
Combine logarithmic terms: Combine the logarithmic terms using the product rule. $\newline$ The product rule states that $\log_b(m) + \log_b(n) = \log_b(m*n)$ , so we can combine the terms. $\newline$ $\log_2(2x \cdot 3^3) = 0$
Simplify expression: Simplify the expression inside the logarithm. $\newline$ $2x \times 3^3 = 2x \times 27$ $\newline$ $\log_2(54x) = 0$
Convert to exponential: Convert the logarithmic equation to an exponential equation. $\newline$ If $\log_2(54x) = 0$ , then $2^0 = 54x$ . $\newline$ $1 = 54x$
Solve for x: Solve for x. $\newline$ Divide both sides by $54$ to isolate $x$ . $\newline$ $x = \frac{1}{54}$