$4 \cdot 5^{-6 t}=2000$ $\newline$ Which of the following is the solution of the equation? $\newline$ Choose $1$ answer: $\newline$ (A) $t=\frac{-\log _{5}(500)}{6}$ $\newline$ (B) $t=\frac{-\log _{20}(2000)}{6}$ $\newline$ (C) $t=\frac{-\log _{2000}(20)}{6}$ $\newline$ (D) $t=\frac{-\log _{500}(5)}{6}$

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Algebra 2

Product property of logarithms

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4 \cdot 5^{-6 t}=2000

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Which of the following is the solution of the equation?

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Choose

1

answer:

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(A)

t=\frac{-\log _{5}(500)}{6}

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(B)

t=\frac{-\log _{20}(2000)}{6}

\newline

(C)

t=\frac{-\log _{2000}(20)}{6}

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(D)

t=\frac{-\log _{500}(5)}{6}

Divide and isolate variable term: Divide both sides of the equation by $4$ to isolate the term with the variable t. $\newline$ Calculation: $\frac{4 \cdot 5^{-6t}}{4} = \frac{2000}{4}$ $\newline$ $5^{-6t} = 500$
Apply logarithm to both sides: Apply the logarithm to both sides of the equation to solve for t. $\newline$ Calculation: $\log(5^{-6t}) = \log(500)$
Use logarithm property: Use the property of logarithms that allows us to bring the exponent in front of the logarithm. $\newline$ Calculation: $-6t \cdot \log(5) = \log(500)$
Divide by $-6$ log( $5$ ): Divide both sides of the equation by $-6 \log(5)$ to solve for t. $\newline$ Calculation: $t = \frac{\log(500)}{-6 \log(5)}$
Correct previous error: Recognize that $\log(500)$ can be simplified to $\log(5^3)$ since $500 = 5^3$ . $\newline$ Calculation: $t = \frac{\log(5^3)}{-6 \log(5)}$
Correct previous error: Recognize that $\log(500)$ can be simplified to $\log(5^3)$ since $500 = 5^3$ . $\newline$ Calculation: $t = \frac{\log(5^3)}{-6 \log(5)}$ Apply the property of logarithms that allows us to bring the exponent out in front of the logarithm. $\newline$ Calculation: $t = \frac{3 \log(5)}{-6 \log(5)}$
Correct previous error: Recognize that $\log(500)$ can be simplified to $\log(5^3)$ since $500 = 5^3$ . $\newline$ Calculation: $t = \frac{\log(5^3)}{-6 \log(5)}$ Apply the property of logarithms that allows us to bring the exponent out in front of the logarithm. $\newline$ Calculation: $t = \frac{3 \log(5)}{-6 \log(5)}$ Simplify the expression by canceling out $\log(5)$ on the numerator and the denominator. $\newline$ Calculation: $t = \frac{3}{-6}$
Correct previous error: Recognize that $\log(500)$ can be simplified to $\log(5^3)$ since $500 = 5^3$ . $\newline$ Calculation: $t = \frac{\log(5^3)}{-6 \log(5)}$ Apply the property of logarithms that allows us to bring the exponent out in front of the logarithm. $\newline$ Calculation: $t = \frac{3 \log(5)}{-6 \log(5)}$ Simplify the expression by canceling out $\log(5)$ on the numerator and the denominator. $\newline$ Calculation: $t = \frac{3}{-6}$ Simplify the fraction to find the value of t. $\newline$ Calculation: $t = -\frac{1}{2}$
Correct previous error: Recognize that $\log(500)$ can be simplified to $\log(5^3)$ since $500 = 5^3$ . $\newline$ Calculation: $t = \frac{\log(5^3)}{-6 \log(5)}$ Apply the property of logarithms that allows us to bring the exponent out in front of the logarithm. $\newline$ Calculation: $t = \frac{3 \log(5)}{-6 \log(5)}$ Simplify the expression by canceling out $\log(5)$ on the numerator and the denominator. $\newline$ Calculation: $t = \frac{3}{-6}$ Simplify the fraction to find the value of t. $\newline$ Calculation: $t = -\frac{1}{2}$ Realize that a mistake was made in the previous steps because the simplification of $\log(500)$ to $\log(5^3)$ was incorrect since $\log(5^3)$ $0$ is not equal to $\log(5^3)$ $1$ . Therefore, we need to correct the error and go back to Step $4$ .