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20*7^(3y)=5
What is the solution of the equation?
Round your answer, if necessary, to the nearest thousandth.

y~~

$20 \cdot 7^{3 y}=5$ $\newline$ What is the solution of the equation? $\newline$ Round your answer, if necessary, to the nearest thousandth. $\newline$ $y \approx$

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Solve exponential equations using common logarithms

Full solution

Q.

20 \cdot 7^{3 y}=5

\newline

What is the solution of the equation?

\newline

Round your answer, if necessary, to the nearest thousandth.

\newline

y \approx

Isolate exponential term: Isolate the exponential term. $\newline$ Divide both sides of the equation by $20$ to isolate the $7^{3y}$ term. $\newline$ $20 \cdot 7^{3y} = 5$ $\newline$ $7^{3y} = \frac{5}{20}$ $\newline$ $7^{3y} = \frac{1}{4}$
Divide by $20$ : Apply the logarithm to both sides of the equation. $\newline$ To solve for $y$ , we can use the natural logarithm ( $\ln$ ) or the common logarithm ( $\log$ ). Here, we'll use the natural logarithm. $\newline$ $\ln(7^{3y}) = \ln(\frac{1}{4})$
Apply logarithm: Use the power property of logarithms. $\newline$ The power property of logarithms states that $\ln(a^b) = b \cdot \ln(a)$ . We apply this property to simplify the left side of the equation. $\newline$ $3y \cdot \ln(7) = \ln\left(\frac{1}{4}\right)$
Use power property: Isolate $y$ . $\newline$ Divide both sides of the equation by $3\ln(7)$ to solve for $y$ . $\newline$ $y = \frac{\ln(\frac{1}{4})}{3\ln(7)}$
Isolate y: Calculate the value of y using a calculator. $\newline$ y = $\frac{\ln(\frac{1}{4})}{3 \cdot \ln(7)}$ $\newline$ y \approx $\frac{-0.317393805}{3 \cdot 1.945910149}$ $\newline$ y \approx $\frac{-0.317393805}{5.837730447}$ $\newline$ y \approx $-0.054355248$ $\newline$ Round the answer to the nearest thousandth. $\newline$ y \approx $-0.054$

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