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Logarithms play a vital role in solving exponential equations. They help us deal with powers by converting multiplication in exponents into addition. Consider the original equation

• \( (1.005)^{12t} = 5 \)

In such equations, logarithms allow us to take the exponent down, making it easier to solve for variables. When you apply the natural logarithm (ln) to both sides, you transform the equation into a manageable linear form. This step is crucial because it transforms the equation into:

• \( \ln((1.005)^{12t}) = \ln(5) \)

Now, the exponent can be brought down using the power rule of logarithms, which states:

• If \( \ln(a^b) = b \cdot \ln(a) \)

So our equation becomes:

• \( 12t \cdot \ln(1.005) = \ln(5) \)

This conversion is a fundamental aspect of logarithms that simplifies solving equations involving exponents.

When solving exponential equations, as shown in the exercise, the ultimate goal is to isolate the variable of interest. Here, the focus is on solving for "t". After applying the logarithm and simplifying, we obtained the equation:

• \( 12t \cdot \ln(1.005) = \ln(5) \)

To isolate "t", we must perform algebraic manipulations:

• Divide both sides by the entire coefficient of "t", which in this case is \( 12 \cdot \ln(1.005) \)

This step rearranges the equation to:

• \( t = \frac{\ln(5)}{12 \cdot \ln(1.005)} \)

This process of isolating the variable is essential because it reveals the solution directly. By understanding how to manipulate and isolate variables, you can solve not just exponential equations, but a wide array of mathematical problems.

Exponentiation is the operation of raising a number to a power, and it is a cornerstone of many mathematical concepts. In the given exercise, the equation involves exponential growth over time, described as:

• \( (1.005)^{12t} = 5 \)

Here, 1.005 represents the base, and \( 12t \) is the exponent, which indicates repeated multiplication of the base.

• The base, 1.005, suggests a rate slightly over 1, implying growth.
• The exponent, 12t, suggests compounding over time, multiplying 12 times "t".

Exponentiation captures the powerful way numbers escalate when multiplied by themselves repeatedly. Understanding this helps you appreciate why logarithms are needed to "undo" exponentiation in equations. It also illustrates how exponential functions can describe phenomena like interest, population growth, and more, where compounding leads to exponential increases over time. By mastering these concepts, you can tackle various real-world issues involving growth and decay.