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

Apply product rule: Apply the product rule of logarithms to combine the logarithmic terms. The product rule of logarithms states that log_b(m) + log_b(n) = log_b(m*n) for the same base b. So, log_4(9x) + log_4(9) becomes log_4(9x * 9).Simplify expression: Simplify the expression inside the logarithm. Multiplying 9x by 9 gives us 81x. So, log_4(9x * 9) becomes log_4(81x).Set equation equal: Set the logarithmic equation equal to 3. We now have log_4(81x) = 3.Convert to exponential form: Convert the logarithmic equation to its exponential form. The exponential form of log_4(81x) = 3 is 4^3 = 81x.Calculate value: Calculate 4^3. 4^3 equals 64. So, the equation becomes 64 = 81x.Solve for x: Solve for x. Divide both sides of the equation by 81 to isolate x. x = 64 / 81.Simplify fraction: Simplify the fraction if possible. 64 and 81 have no common factors other than 1, so the fraction is already in its simplest form. x = 64 / 81.

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Solve for the exact value of
x.

log_(4)(9x)+log_(4)(9)=3
Answer:

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

\newline

Answer:

Apply product rule: Apply the product rule of logarithms to combine the logarithmic terms. $\newline$ The product rule of logarithms states that $\log_b(m) + \log_b(n) = \log_b(m*n)$ for the same base $b$ . $\newline$ So, $\log_4(9x) + \log_4(9)$ becomes $\log_4(9x \times 9)$ .
Simplify expression: Simplify the expression inside the logarithm. $\newline$ Multiplying $9x$ by $9$ gives us $81x$ . $\newline$ So, $\log_4(9x \times 9)$ becomes $\log_4(81x)$ .
Set equation equal: Set the logarithmic equation equal to $3$ . We now have $\log_4(81x) = 3$ .
Convert to exponential form: Convert the logarithmic equation to its exponential form. $\newline$ The exponential form of $\log_4(81x) = 3$ is $4^3 = 81x$ .
Calculate value: Calculate $4^3$ . $4^3$ equals $64$ . So, the equation becomes $64 = 81x$ .
Solve for x: Solve for x. $\newline$ Divide both sides of the equation by $81$ to isolate $x$ . $\newline$ $x = \frac{64}{81}$ .
Simplify fraction: Simplify the fraction if possible. $64$ and $81$ have no common factors other than $1$ , so the fraction is already in its simplest form. $x = \frac{64}{81}$ .