Solve for the exact value of x. log_(4)(2x)-3log_(4)(9)=0 Answer:

Apply Power Rule: Let's start by applying the power rule of logarithms to the term with the coefficient of 3. Power rule of logarithms: log_b(a^n) = n * log_b(a) 3log_4(9) = log_4(9^3)Rewrite Equation: Now, let's rewrite the equation using the result from the previous step. log_4(2x) - log_4(9^3) = 0Combine Logarithmic Expressions: Next, we can apply the quotient rule of logarithms to combine the two logarithmic expressions into a single logarithm. Quotient rule of logarithms: log_b(a) - log_b(c) = log_b(a/c) log_4(2x/9^3) = 0Rewrite in Exponential Form: To solve for x, we need to rewrite the logarithmic equation in exponential form. If log_b(a) = c, then b^c = a 4^0 = 2x/9^3Solve for x: Since 4^0 is equal to 1, we can now solve for x. 1 = 2x/9^3Isolate x: Multiply both sides of the equation by 9^3 to isolate x on one side. 9^3 = 2xDivide by 2: Now, divide both sides by 2 to solve for x. x = 9^3 / 2Calculate x: Calculate the value of 9^3 and then divide by 2. 9^3 = 729 x = 729 / 2Finally, we get the exact value of x. x = 364.5

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Solve for the exact value of
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log_(4)(2x)-3log_(4)(9)=0
Answer:

Solve for the exact value of $x$ . $\newline$ $\log _{4}(2 x)-3 \log _{4}(9)=0$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Solve for the exact value of

x

\newline

\log _{4}(2 x)-3 \log _{4}(9)=0

\newline

Answer:

Apply Power Rule: Let's start by applying the power rule of logarithms to the term with the coefficient of $3$ . $\newline$ Power rule of logarithms: $\log_b(a^n) = n \cdot \log_b(a)$ $\newline$ $3\log_4(9) = \log_4(9^3)$
Rewrite Equation: Now, let's rewrite the equation using the result from the previous step. $\log_4(2x) - \log_4(9^3) = 0$
Combine Logarithmic Expressions: Next, we can apply the quotient rule of logarithms to combine the two logarithmic expressions into a single logarithm. $\newline$ Quotient rule of logarithms: $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$ $\newline$ $\log_4\left(\frac{2x}{9^3}\right) = 0$
Rewrite in Exponential Form: To solve for $x$ , we need to rewrite the logarithmic equation in exponential form. $\newline$ If $\log_b(a) = c$ , then $b^c = a$ $\newline$ $4^0 = \frac{2x}{9^3}$
Solve for x: Since $4^0$ is equal to $1$ , we can now solve for $x$ . $\newline$ $1 = \frac{2x}{9^3}$
Isolate $x$ : Multiply both sides of the equation by $9^3$ to isolate $x$ on one side. $\newline$ $9^3 = 2x$
Divide by $2$ : Now, divide both sides by $2$ to solve for x. $\newline$ $x = \frac{9^3}{2}$
Calculate $x$ : Calculate the value of $9^3$ and then divide by $2$ . $9^3 = 729$ $x = \frac{729}{2}$
Calculate x: Calculate the value of $9^3$ and then divide by $2$ . $\newline$ $9^3 = 729$ $\newline$ $x = 729 / 2$ Finally, we get the exact value of $x$ . $\newline$ $x = 364.5$