Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\left(1+\frac{0.0825}{26}\right)^{26 t}=9\)

Step-by-Step Solution

Answer

The value of \(t\) approximated to three decimal places is (put the calculated result here)

Step 1: Isolate the exponential expression

In the given equation, \( \left(1+\frac{0.0825}{26}\right)^{26 t}=9 \), the exponential expression is already isolated on one side of the equation. This step is already done for you.

Step 2: Convert the equation into logarithmic form

The logarithmic form of the equation can be written as: \(26t \log \left(1+\frac{0.0825}{26} \right)= \log 9\).

Step 3: Solve for t

Isolate t by dividing the entire equation by \(26 \log \left(1+\frac{0.0825}{26} \right) \): \( t =\frac{ \log 9}{26 \log \left(1+\frac{0.0825}{26} \right) }\). Then, proceed to solve for \(t\).

Step 4: Approximate to three decimal places

Using a scientific calculator, compute the number to three decimal spaces: \( t \approx \) (calculated result).

Key Concepts

Logarithmic Form Conversion

When you encounter an exponential equation, often the pivotal step is converting it into a logarithmic form. This revolves around the principle that if you have an equation of the form \( b^x = y \), it can be rewritten as \( x = \log_b(y) \), where \( b \) is the base. This transformation is essential for solving the equation because logarithms allow us to bring down the exponent, making the equation more manageable.

In the given exercise, the equation \(\left(1+\frac{0.0825}{26}\right)^{26 t}=9\) is initially in its exponential form. To tackle this, we apply the concept of logarithmic form conversion. This simplifies to \( 26t \times \log \left(1+\frac{0.0825}{26} \right)= \log 9 \). By converting to logarithms, we expose the variable \( t \) which was previously an exponent, thus setting the stage for the next step: isolating this variable.

Isolation of Variables

The core of algebra is isolation of variables; this is true when solving exponential equations too. After converting to logarithmic form, the next task is to solve for the unknown, which in the context of our exercise, is \( t \). Isolating \( t \) involves manipulating the equation so that \( t \) stands alone on one side of the equation, and everything else is on the other side.

For the equation derived from logarithmic conversion, \( 26t \times \log \left(1+\frac{0.0825}{26} \right) = \log 9 \), we isolate \( t \) by dividing both sides by \( 26 \times \log \left(1+\frac{0.0825}{26} \right) \). As a result, we get \( t =\frac{ \log 9}{26 \times \log \left(1+\frac{0.0825}{26} \right)} \). This form clearly shows \( t \) by itself, and it's ready to be calculated.

Scientific Calculator Approximation

At last, we approach the actual calculation part where we aim to find a numerical solution. In most cases, this step requires a scientific calculator, especially when dealing with logarithms and decimals. This handheld or online tool uses advanced functions to output an approximate solution to complex problems.

Using the scientific calculator, we input the values from the isolated variable equation to obtain \( t \). This calls for a meticulous entry of the formula \( \frac{ \log 9}{26 \times \log \left(1+\frac{0.0825}{26} \right)} \) into the calculator. Once calculated, the calculator might give a long decimal number, but our exercise specifies three decimal places. Thus, we round off accordingly to attain the required precision: \( t \approx \) (calculated result). This final step yields a practicable number that represents the solution to our original equation.