Question

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

Explanation:

Step1: Isolate the exponential term

Divide both sides by 18:
$\frac{18 \cdot 2^{5t}}{18} = \frac{261}{18}$
Simplify to get:
$2^{5t} = 14.5$

Step2: Apply natural logarithm

Take $\ln$ of both sides:
$\ln(2^{5t}) = \ln(14.5)$
Use log power rule $\ln(a^b)=b\ln(a)$:
$5t \cdot \ln(2) = \ln(14.5)$

Step3: Solve for t

Isolate t:
$t = \frac{\ln(14.5)}{5\ln(2)}$

Step4: Calculate the value

Substitute $\ln(14.5) \approx 2.67414864$ and $\ln(2) \approx 0.69314718$:
$t \approx \frac{2.67414864}{5 \cdot 0.69314718} = \frac{2.67414864}{3.4657359} \approx 0.772$

Answer:

$t \approx 0.772$