Solve for the exact value of x. log_(4)(9x)-2log_(4)(7)=1 Answer:

Apply power rule: Apply the power rule of logarithms to the term with the coefficient. The power rule states that a*log_b(x) = log_b(x^a). We apply this to the second term of the equation. log_4(9x) - log_4(7^2) = 1Combine using quotient rule: Combine the logarithmic expressions using the quotient rule. The quotient rule states that log_b(x) - log_b(y) = log_b(x/y). We apply this to combine the two logarithmic expressions. log_4(9x/49) = 1Convert to exponential form: Convert the logarithmic equation to its exponential form. The exponential form of log_b(x) = y is b^y = x. We apply this to solve for x. 4^1 = 9x/49Solve for x: Solve for x. Multiply both sides of the equation by 49 to isolate x. 49 * 4 = 9x 196 = 9xDivide to find x: Divide both sides by 9 to find the value of x. x = 196 / 9 x = 21.7777777778

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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 _{4}(9 x)-2 \log _{4}(7)=1

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Answer:

Apply power rule: Apply the power rule of logarithms to the term with the coefficient. $\newline$ The power rule states that $a\log_b(x) = \log_b(x^a)$ . We apply this to the second term of the equation. $\newline$ $\log_4(9x) - \log_4(7^2) = 1$
Combine using quotient rule: Combine the logarithmic expressions using the quotient rule. $\newline$ The quotient rule states that $\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$ . We apply this to combine the two logarithmic expressions. $\newline$ $\log_4\left(\frac{9x}{49}\right) = 1$
Convert to exponential form: Convert the logarithmic equation to its exponential form. $\newline$ The exponential form of $\log_b(x) = y$ is $b^y = x$ . We apply this to solve for $x$ . $\newline$ $4^1 = \frac{9x}{49}$
Solve for x: Solve for x. $\newline$ Multiply both sides of the equation by $49$ to isolate x. $\newline$ $49 \times 4 = 9x$ $\newline$ $196 = 9x$
Divide to find x: Divide both sides by $9$ to find the value of x. $\newline$ $x = \frac{196}{9}$ $\newline$ $x = 21.7777777778$