Math  /  Algebra

QuestionSolve the equation; enter your answer rounded to four decimal places. 14e9t=8+2e9t14 e^{9 t}=8+2 e^{9 t} One solution \square No solution

Studdy Solution

STEP 1

1. The equation 14e9t=8+2e9t 14 e^{9t} = 8 + 2 e^{9t} is an exponential equation.
2. We need to isolate the exponential term and solve for t t .
3. The solution should be rounded to four decimal places.

STEP 2

1. Isolate the exponential term e9t e^{9t} .
2. Solve for e9t e^{9t} .
3. Take the natural logarithm to solve for t t .
4. Round the solution to four decimal places.

STEP 3

First, subtract 2e9t 2 e^{9t} from both sides to isolate the exponential term:
14e9t2e9t=8 14 e^{9t} - 2 e^{9t} = 8 12e9t=8 12 e^{9t} = 8

STEP 4

Divide both sides by 12 to solve for e9t e^{9t} :
e9t=812 e^{9t} = \frac{8}{12} e9t=23 e^{9t} = \frac{2}{3}

STEP 5

Take the natural logarithm of both sides to solve for 9t 9t :
ln(e9t)=ln(23) \ln(e^{9t}) = \ln\left(\frac{2}{3}\right) 9t=ln(23) 9t = \ln\left(\frac{2}{3}\right)
Now, divide both sides by 9 to solve for t t :
t=ln(23)9 t = \frac{\ln\left(\frac{2}{3}\right)}{9}

STEP 6

Calculate the value of t t and round to four decimal places:
First, compute ln(23) \ln\left(\frac{2}{3}\right) :
ln(23)0.405465 \ln\left(\frac{2}{3}\right) \approx -0.405465
Now, divide by 9:
t0.4054659 t \approx \frac{-0.405465}{9} t0.0450517 t \approx -0.0450517
Round to four decimal places:
t0.0451 t \approx -0.0451
The solution is:
0.0451 \boxed{-0.0451}

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