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Chapter 3: Problem 43

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$\left(1+\frac{0.065}{365}\right)^{365 t}=4$$

Short Answer

Expert verified
To solve the given exponential equation, we isolated \(t\) and used logarithm properties to make the calculation simpler. By inserting the given values into a calculator, we find that \(t\) approximates to 10.536 when rounded to three decimal places.

Step by step solution

01

Write down the given equation

Write down the equation: \((1+0.065/365)^{365 t} = 4\)
02

Take natural log on both sides

To isolate \(t\), we need to eliminate the exponent. This can be achieved by using the logarithm property. Apply natural log (ln) on both sides: \(\ln((1+0.065/365)^{365 t}) = \ln(4) \).
03

Use logarithm property

Use the logarithm property that states \(\ln(a^b) = b\ln(a)\). So, it should look like this : \( 365t \cdot \ln(1+0.065/365) = \ln(4) \).
04

Isolate \(t\)

Dividing both sides of the equation by \(365 \cdot \ln(1+0.065/365)\) to isolate \(t\), we get: \( t = \frac{\ln(4)}{365 \cdot \ln(1+0.065/365)} \)
05

Compute \(t\)

Insert the values into a calculator to get the value of \(t\). Note that we are asked to get the solution to three decimal places.

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