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Chapter 3: Problem 44

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$\left(1+\frac{0.10}{12}\right)^{12 t}=2$$

Short Answer

Expert verified
The solution to the equation, when approximated to three decimal places, is \(t \approx 6.928\).

Step by step solution

01

Rewrite the equation in exponential form.

The given equation can be rewritten without the brackets for simplification: \((1+ \frac{0.10}{12})^{12t} = 2\).
02

Apply Natural Logarithm (ln) to Both Sides

Applying natural logarithm to both sides gives: \(\ln((1+ \frac{0.10}{12})^{12t}) = \ln(2)\). Using the logarithmic law that allows us to bring down the exponent, we arrive at: \(12t \cdot \ln(1+ \frac{0.10}{12}) = \ln(2)\).
03

Solve For t

Solving the equation for \(t\) we get: \(t = \frac{\ln(2)}{12\cdot \ln(1+\frac{0.10}{12})}\).
04

Use Calculator for Approximation

By plugging into a calculator, approximate \(t\) up to three decimal places.

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