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

Apply power rule: Apply the power rule of logarithms to simplify the term with the coefficient. The power rule of logarithms states that n*log_b(a) = log_b(a^n). Let's apply this to the term 4log_4(3). 4log_4(3) = log_4(3^4)Calculate value: Calculate the value of 3^4. 3^4 = 3 * 3 * 3 * 3 = 81Rewrite equation: Rewrite the equation using the result from Step 2. log_4(9x) - log_4(81) = 0Apply quotient rule: Apply the quotient rule of logarithms to combine the logarithms. The quotient rule of logarithms states that log_b(a) - log_b(c) = log_b(a/c). Let's apply this to combine the logarithms. log_4(9x/81) = 0Simplify fraction: Simplify the fraction inside the logarithm. 9x/81 = x/9 So, log_4(x/9) = 0Convert to exponential: Convert the logarithmic equation to an exponential equation. If log_b(a) = c, then b^c = a. Let's apply this to solve for x. 4^0 = x/9Calculate value: Calculate the value of 4^0. 4^0 = 1Solve for x: Solve for x. x/9 = 1 x = 9

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Solve for the exact value of
x.

log_(4)(9x)-4log_(4)(3)=0
Answer:

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

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Math Problems

Precalculus

Quotient property of logarithms

Full solution

Q. Solve for the exact value of

x

\newline

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

\newline

Answer:

Apply power rule: Apply the power rule of logarithms to simplify the term with the coefficient. $\newline$ The power rule of logarithms states that $n\log_b(a) = \log_b(a^n)$ . Let's apply this to the term $4\log_4(3)$ . $\newline$ $4\log_4(3) = \log_4(3^4)$
Calculate value: Calculate the value of $3^4$ . $\newline$ $3^4 = 3 \times 3 \times 3 \times 3 = 81$
Rewrite equation: Rewrite the equation using the result from Step $2$ . $\newline$ $\log_4(9x) - \log_4(81) = 0$
Apply quotient rule: Apply the quotient rule of logarithms to combine the logarithms. $\newline$ The quotient rule of logarithms states that $\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)$ . Let's apply this to combine the logarithms. $\newline$ $\log_4\left(\frac{9x}{81}\right) = 0$
Simplify fraction: Simplify the fraction inside the logarithm. $\newline$ $\frac{9x}{81} = \frac{x}{9}$ $\newline$ So, $\log_4\left(\frac{x}{9}\right) = 0$
Convert to exponential: Convert the logarithmic equation to an exponential equation. $\newline$ If $\log_b(a) = c$ , then $b^c = a$ . Let's apply this to solve for $x$ . $\newline$ $4^0 = \frac{x}{9}$
Calculate value: Calculate the value of $4^0$ . $\newline$ $4^0 = 1$
Solve for x: Solve for x. $\newline$ $\frac{x}{9} = 1$ $\newline$ $x = 9$