Solve for the exact value of x. log_(7)(5x)-3log_(7)(2)=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 3log_7(2). Power rule application: 3log_7(2) = log_7(2^3) Calculation: 2^3 = 8Rewrite Equation: Rewrite the original equation using the result from Step 1. The original equation log_7(5x) - 3log_7(2) = 0 becomes log_7(5x) - log_7(8) = 0.Apply Quotient Rule: Apply the quotient rule of logarithms to combine the logarithms into a single logarithm. 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. Quotient rule application: log_7(5x) - log_7(8) = log_7(5x/8)Set Equal to Base: Set the argument of the logarithm equal to the base raised to the power of the other side of the equation. Since log_7(5x/8) = 0, we can write this as 7^0 = 5x/8. Calculation: 7^0 = 1Solve for x: Solve for x. We have the equation 1 = 5x/8. Multiply both sides by 8 to isolate x. Calculation: 8 * 1 = 5xComplete Calculation: Complete the calculation to find the value of x. Calculation: 8 = 5x Divide both sides by 5 to solve for x. Calculation: x = 8/5

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Solve for the exact value of
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log_(7)(5x)-3log_(7)(2)=0
Answer:

Solve for the exact value of $x$ . $\newline$ $\log _{7}(5 x)-3 \log _{7}(2)=0$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Solve for the exact value of

x

\newline

\log _{7}(5 x)-3 \log _{7}(2)=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 $3\log_7(2)$ . $\newline$ Power rule application: $3\log_7(2) = \log_7(2^3)$ $\newline$ Calculation: $2^3 = 8$
Rewrite Equation: Rewrite the original equation using the result from Step $1$ . $\newline$ The original equation $\log_7(5x) - 3\log_7(2) = 0$ becomes $\log_7(5x) - \log_7(8) = 0$ .
Apply Quotient Rule: Apply the quotient rule of logarithms to combine the logarithms into a single logarithm. $\newline$ 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. $\newline$ Quotient rule application: $\log_7(5x) - \log_7(8) = \log_7\left(\frac{5x}{8}\right)$
Set Equal to Base: Set the argument of the logarithm equal to the base raised to the power of the other side of the equation. $\newline$ Since $\log_7(\frac{5x}{8}) = 0$ , we can write this as $7^0 = \frac{5x}{8}$ . $\newline$ Calculation: $7^0 = 1$
Solve for x: Solve for x. $\newline$ We have the equation $1 = \frac{5x}{8}$ . Multiply both sides by $8$ to isolate $x$ . $\newline$ Calculation: $8 \times 1 = 5x$
Complete Calculation: Complete the calculation to find the value of $x$ . $\newline$ Calculation: $8 = 5x$ $\newline$ Divide both sides by $5$ to solve for $x$ . $\newline$ Calculation: $x = \frac{8}{5}$