Solve for the exact value of x. log_(5)(7x)+2log_(5)(5)=4 Answer:

Apply Power Rule: Apply the power rule of logarithms to simplify the second term. The power rule of logarithms states that log_b(a^c) = c * log_b(a). 2log_5(5) can be rewritten as log_5(5^2). Calculation: log_5(5^2) = log_5(25).Substitute Simplified Term: Substitute the simplified second term back into the equation. The equation now becomes log_5(7x) + log_5(25) = 4.Apply Product Rule: Apply the product rule of logarithms to combine the two logarithmic terms. The product rule states that log_b(a) + log_b(c) = log_b(a * c). Combine the terms: log_5(7x) + log_5(25) = log_5(7x * 25). Calculation: log_5(175x) = 4.Convert to Exponential Form: Convert the logarithmic equation to its exponential form. The exponential form of log_b(a) = c is b^c = a. Convert the equation: 5^4 = 175x. Calculation: 625 = 175x.Solve for x: Solve for x. Divide both sides of the equation by 175 to isolate x. Calculation: x = 625 / 175. Calculation: x = 3.57142857143 (This is a repeating decimal, so we can express it as a fraction). Calculation: x = 25/7.

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Solve for the exact value of
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log_(5)(7x)+2log_(5)(5)=4
Answer:

Solve for the exact value of $x$ . $\newline$ $\log _{5}(7 x)+2 \log _{5}(5)=4$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Solve for the exact value of

x

\newline

\log _{5}(7 x)+2 \log _{5}(5)=4

\newline

Answer:

Apply Power Rule: Apply the power rule of logarithms to simplify the second term. $\newline$ The power rule of logarithms states that $\log_b(a^c) = c \cdot \log_b(a)$ . $\newline$ $2\log_5(5)$ can be rewritten as $\log_5(5^2)$ . $\newline$ Calculation: $\log_5(5^2) = \log_5(25)$ .
Substitute Simplified Term: Substitute the simplified second term back into the equation. $\newline$ The equation now becomes $\log_5(7x) + \log_5(25) = 4$ .
Apply Product Rule: Apply the product rule of logarithms to combine the two logarithmic terms. $\newline$ The product rule states that $\log_b(a) + \log_b(c) = \log_b(a \cdot c)$ . $\newline$ Combine the terms: $\log_5(7x) + \log_5(25) = \log_5(7x \cdot 25)$ . $\newline$ Calculation: $\log_5(175x) = 4$ .
Convert to Exponential Form: Convert the logarithmic equation to its exponential form. $\newline$ The exponential form of $\log_b(a) = c$ is $b^c = a$ . $\newline$ Convert the equation: $5^4 = 175x$ . $\newline$ Calculation: $625 = 175x$ .
Solve for x: Solve for x. $\newline$ Divide both sides of the equation by $175$ to isolate $x$ . $\newline$ Calculation: $x = \frac{625}{175}$ . $\newline$ Calculation: $x = 3.57142857143$ (This is a repeating decimal, so we can express it as a fraction). $\newline$ Calculation: $x = \frac{25}{7}$ .