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18*2^(5t)=261
What is the solution of the equation?
Round your answer, if necessary, to the nearest thousandth.

t~~

$18 \cdot 2^{5 t}=261$ $\newline$ What is the solution of the equation? $\newline$ Round your answer, if necessary, to the nearest thousandth. $\newline$ $t \approx$

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Solve exponential equations using common logarithms

Full solution

Q.

18 \cdot 2^{5 t}=261

\newline

What is the solution of the equation?

\newline

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

\newline

t \approx

Write and isolate exponential term: Write down the given equation and isolate the exponential term. $\newline$ We have the equation $18 \cdot 2^{5t} = 261$ . To isolate the exponential term, we need to divide both sides of the equation by $18$ . $\newline$ $\frac{261}{18} = 2^{5t}$
Calculate division result: Calculate the result of the division on the left side of the equation. $\newline$ $261 / 18 = 14.5$ $\newline$ So, we have $14.5 = 2^{5t}$ .
Apply logarithm to both sides: Apply the logarithm to both sides of the equation to solve for $t$ . We can use the natural logarithm ( $\ln$ ) for this purpose. $\ln(14.5) = \ln(2^{5t})$
Use power property of logarithms: Use the power property of logarithms to bring down the exponent. $\newline$ The power property states that $\ln(a^b) = b \cdot \ln(a)$ . $\newline$ $\ln(14.5) = 5t \cdot \ln(2)$
Isolate $t$ by dividing: Isolate $t$ by dividing both sides of the equation by $5$ \cdot \ln( $2$ ). $\newline$ $t = \frac{\ln($ $14$ . $5$ )}{ $5$ \cdot \ln( $2$ )}
Calculate value of t: Calculate the value of t using a calculator. $\newline$ t = $\frac{\ln(14.5)}{5 \cdot \ln(2)}$ $\newline$ t \approx $\frac{1.273}{5 \cdot 0.693}$ $\newline$ t \approx $\frac{1.273}{3.465}$ $\newline$ t \approx $0.367$

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