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Understanding how to rewrite exponential equations in logarithmic form is crucial for solving them algebraically. To transform an equation such as \( 5^{-t / 2} = 0.20 \) into logarithmic form, we exploit the definition of a logarithm. A logarithm answers the question: to what power must the base be raised to produce a given number? In our example, we are seeking the power to which 5 must be raised to result in 0.20.

So, we express \( 5^{-t / 2} \) as \( \log_{5}{0.20} \). The equation becomes \( -t / 2 = \log_{5}{0.20} \), which is the logarithmic form. This form is particularly useful because it brings the variable \( t \) out of the exponent, making it solvable through regular algebraic methods.

Steps for Converting to Logarithmic Form

• Identify the base of the exponential expression (in this case, 5).
• Determine the number you are comparing it to or equating it with (0.20).
• Use the logarithmic notation with the base identified in step 1 and equate it to the exponent of the base in the original equation.

Once in logarithmic form, the solution process becomes more straightforward and manageable.

After finding the solution algebraically, it's important to verify that the solution is correct. A graphing utility is an invaluable tool for this verification process. By inputting the original exponential equation into a graphing utility, one can plot the graph and visually confirm the solution.

For the equation \( 5^{-t / 2} = 0.20 \), you would input \( y = 5^{-t / 2} \) into the graphing utility and observe where the graph intersects the horizontal line \( y = 0.20 \). The x-value at this intersection point should correspond to your calculated value of \( t \).

Benefits of Using a Graphing Utility

• Provides visual confirmation of the solution's accuracy.
• Helps in understanding the behavior of the exponential function.
• Acts as a quick check against possible calculation errors.

If the graph and the algebraic solution align, you can be confident in the correctness of your solution.

When rounding your result to three decimal places, as required by the exercise, you ensure that your answer is accurate to a specific level of precision, which is particularly important in scientific calculations. To solve the equation \( -t / 2 = \log_{5}{0.20} \) algebraically, you need to isolate the variable \( t \) by performing inverse operations.

The first step is to multiply both sides by -2 to cancel out the division by 2 on the left side, resulting in \( t = -2 \cdot \log_{5}{0.20} \). This form of the equation allows you to use a calculator to compute the logarithm to the base 5 of 0.20 and find the value of \( t \).

Key Algebraic Steps:

• Multiplying or dividing to isolate the variable term.
• Applying specific logarithm rules or using a calculator for non-standard bases.
• Rounding to the required number of decimal places for precision.

Mastering these algebraic manipulations is essential for solving exponential equations efficiently.