Question

question 9

if (e^{4x} = 15), then (x = )

question 10

find the solution of the exponential equation
(1000(1.06)^{2t} = 50,000)
in terms of logarithms, or correct to four decimal places.
(t = )

Explanation:

⚡ Using what you learned: Solving Exponential Equations

Step 1: Solve Question 9

Take the natural logarithm (\(\ln\)) of both sides:

$$ \ln(e^{4x}) = \ln(15) $$

Apply the property \(\ln(e^u) = u\):

$$ 4x = \ln(15) $$

Divide by \(4\):

$$ x = \frac{\ln(15)}{4} $$

Step 2: Solve Question 10

Isolate the exponential term by dividing both sides by \(1000\):

$$ (1.06)^{2t} = 50 $$

Take the natural logarithm of both sides:

$$ \ln((1.06)^{2t}) = \ln(50) $$

Bring the exponent to the front:

$$ 2t \ln(1.06) = \ln(50) $$

Solve for \(t\):

$$ t = \frac{\ln(50)}{2 \ln(1.06)} $$

Calculate the decimal value:

$$ t \approx \frac{3.912023}{2 \times 0.0582689} \approx 33.5685 $$

Answer:

Question 9:

$$ x = \frac{\ln(15)}{4} \approx 0.6770 $$

Question 10:

$$ t = \frac{\ln(50)}{2\ln(1.06)} \approx 33.5685 $$