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2*3^((2x)/(7))=30
Which of the following is the solution of the equation?
Choose 1 answer:
(A)
x=(7)/(2)log_(3)(15)
(B)
x=(7)/(2)log_(15)(3)
(C)
x=(7)/(2)log_(30)(6)
(D)
x=(7)/(2)log_(6)(30)

$2 \cdot 3^{\frac{2 x}{7}}=30$ $\newline$ Which of the following is the solution of the equation? $\newline$ Choose $1$ answer: $\newline$ (A) $x=\frac{7}{2} \log _{3}(15)$ $\newline$ (B) $x=\frac{7}{2} \log _{15}(3)$ $\newline$ C) $x=\frac{7}{2} \log _{30}(6)$ $\newline$ (D) $x=\frac{7}{2} \log _{6}(30)$

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Product property of logarithms

Full solution

Q.

2 \cdot 3^{\frac{2 x}{7}}=30

\newline

Which of the following is the solution of the equation?

\newline

Choose

1

answer:

\newline

(A)

x=\frac{7}{2} \log _{3}(15)

\newline

(B)

x=\frac{7}{2} \log _{15}(3)

\newline

C)

x=\frac{7}{2} \log _{30}(6)

\newline

(D)

x=\frac{7}{2} \log _{6}(30)

Divide and isolate exponential term: Divide both sides of the equation by $2$ to isolate the exponential term. $\newline$ $\frac{2 \cdot 3^{\frac{2x}{7}}}{2} = \frac{30}{2}$ $\newline$ $3^{\frac{2x}{7}} = 15$
Apply logarithm to both sides: Apply the logarithm to both sides of the equation to solve for the exponent. We will use the natural logarithm for convenience, but any logarithm base can be used. $\newline$ $\ln(3^{\frac{2x}{7}}) = \ln(15)$
Use power rule of logarithms: Use the power rule of logarithms to bring down the exponent in front of the logarithm. $\newline$ $\frac{2x}{7} \cdot \ln(3) = \ln(15)$
Isolate x by multiplying both sides: Isolate x by multiplying both sides of the equation by $\frac{7}{2\ln(3)}$ . $\newline$ $x = \frac{7}{2\ln(3)} \cdot \ln(15)$
Recognize equivalence with change of base formula: Recognize that $\frac{\ln(15)}{\ln(3)}$ is equivalent to $\log_3(15)$ by the change of base formula for logarithms. $\newline$ $x = \frac{7}{2} \cdot \log_3(15)$

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