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Math  /  Algebra

QuestionFind the solution of the exponential equation 100(1.04)2t=500,000100(1.04)^{2 t}=500,000 in terms of logarithms, or correct to four decimal places. t=t= □\square Quest □\square Funcs Trig □\square x□x_{\square} \sqrt{ } n\sqrt[n]{ } □\square ↑\uparrow 1

Studdy Solution

STEP 1

1. The equation 100(1.04)2t=500,000 100(1.04)^{2t} = 500,000 is an exponential equation.
2. We will use logarithms to solve for the variable t t .

STEP 2

1. Isolate the exponential expression.
2. Apply logarithms to both sides.
3. Solve for t t .

STEP 3

First, divide both sides of the equation by 100 to isolate the exponential expression:
100(1.04)2t=500,000 100(1.04)^{2t} = 500,000 (1.04)2t=500,000100 (1.04)^{2t} = \frac{500,000}{100} (1.04)2t=5,000 (1.04)^{2t} = 5,000

STEP 4

Apply the logarithm to both sides of the equation. We can use the natural logarithm (ln) or the common logarithm (log), but for this solution, we'll use the natural logarithm:
ln⁡((1.04)2t)=ln⁡(5,000) \ln((1.04)^{2t}) = \ln(5,000)

STEP 5

Use the logarithmic identity ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b \cdot \ln(a) to bring down the exponent:
2t⋅ln⁡(1.04)=ln⁡(5,000) 2t \cdot \ln(1.04) = \ln(5,000)

STEP 6

Solve for t t by dividing both sides by 2⋅ln⁡(1.04) 2 \cdot \ln(1.04) :
2t=ln⁡(5,000)ln⁡(1.04) 2t = \frac{\ln(5,000)}{\ln(1.04)} t=ln⁡(5,000)2⋅ln⁡(1.04) t = \frac{\ln(5,000)}{2 \cdot \ln(1.04)}
Calculate the value of t t to four decimal places:
t≈ln⁡(5,000)2⋅ln⁡(1.04)≈8.51720.0781≈54.5932 t \approx \frac{\ln(5,000)}{2 \cdot \ln(1.04)} \approx \frac{8.5172}{0.0781} \approx 54.5932
The value of t t is approximately:
54.5932 \boxed{54.5932}

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