Question

In Exercises, solve for \(x\) or \(t\). $$ \left(1+\frac{0.07}{12}\right)^{12 t}=3 $$

Short answer

The solution for \( t \) in the equation \((1+0.07/12)^{12t} = 3\) is \[ t = \frac{ln(3)}{12*ln(1+\frac{0.07}{12})} \]

Step-by-step solution

Understand The Equation

This equation resembles the equation of compound interest which is of the form \( (1 + r/n)^{nt} = A \) where r is the interest rate, n is the number of times the interest is compounded per period, t is the time the money is invested for, and A is the amount of money accumulated after n years. Here A = 3, r/n = 0.07/12 and we are solving for t.

Solving For The Exponent

First isolate the exponential expression on one side of the equation and then take the natural logarithm (ln) of both sides of the equation: \[ ln((1+\frac{0.07}{12})^{12t}) = ln(3) \]

Applying The Power Rule Of Logarithms

By using the power rule for logarithms, bring down the exponent (12t) to simplify the equation further: \[ 12t*ln(1+\frac{0.07}{12}) = ln(3) \] The power rule equation is \[ log_b(m^n) = n*log_b(m) \]. This rule allows us to pull t out of the logarithm.

Final Step: Solve For \( t \)

Finally divide both sides by \[ 12*ln(1+\frac{0.07}{12}) \] to get: \[ t = \frac{ln(3)}{12*ln(1+\frac{0.07}{12})} \]

Key concepts

Compound Interest Formula

To understand the compound interest formula, consider the following principle: money invested will grow over time by adding interest to the initial sum — and, crucially, future interest payments are calculated on the total amount, which includes the previously accumulated interest. This is the essence of compounding. The compound interest formula is represented as
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]
where:

• \(A\) is the future value of the investment/loan, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time the money is invested or borrowed for, in years.

In the given exercise, we have the compound interest formula with a specific interest rate and number of compounding periods, and we aim to find the time \(t\) it takes for the initial investment to grow to three times its original size.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number is the power to which \(e\) must be raised to obtain that number. For instance, \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\), representing the inverse nature of exponential functions and logarithms.

Working with natural logarithms is particularly useful when dealing with exponential growth or decay processes, such as compound interest. By applying the natural logarithm to both sides of the equation in the outlined exercise, we are able to transform the exponential equation into a linear form, this brings us one step closer to isolating the variable \(t\) and solving for it.

Power Rule of Logarithms

The power rule of logarithms is a critical tool for simplifying the calculation of logarithms when the argument is raised to a power. It states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number:
\[\log_b(m^n) = n \cdot \log_b(m)\]
This feature is extremely useful, as it allows us to manipulate the positioning of the exponent and hence make complex equations more tractable. In the context of the exercise, we apply this power rule to move the time variable \(t\), which was originally the exponent in the compound interest formula, in front of the logarithm of the base. This effectively changes the equation from an exponential to a linear one, enabling us to solve for \(t\) through relatively simple algebraic operations. Using this rule transforms the seemingly daunting task of dealing with exponents to one where standard algebraic techniques can be applied to find the desired solution.