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Logarithmic functions are the inverse of exponential functions. In basic terms, if you have an exponential function where a number is raised to a certain power to achieve another number, a logarithmic function tells you the power that the original number needs to be raised to, to get that other number. This concept is best expressed by the equation:
\[ b^y = x \Rightarrow \log_b(x) = y \]
Here, \( b \) is the base of the logarithm, \( y \) is the logarithm (the power), and \( x \) is the result you get when raising \( b \) to the power of \( y \). When you see an equation like \( \ln x = 1.6 \), it involves the natural logarithm, which is a specific type of logarithmic function where the base is \( e \), the Euler's number (approximately equal to 2.718).

To solve logarithmic equations like the given exercise, one must understand the relationship between logarithms and exponentiation. The statement \( \ln x = 1.6 \) signifies that the power you need to raise \( e \) to get \( x \) is 1.6. It's crucial to grasp this connection to correctly approach and solve logarithmic equations.

The natural logarithm, denoted as \( \ln \), has the Euler's number \( e \) as its base, which is an irrational and transcendental number approximately equal to 2.71828. It plays a significant role in various branches of mathematics, especially in solving problems involving growth and decay, such as in finance with compound interest or in biology with population models.

Using the natural logarithm, the equation \( \ln x \) equates to asking, 'To what power must \( e \) be raised to obtain \( x \)?' For the exercise provided, \( \ln x = 1.6 \) simplifies to \( e^{1.6} = x \), which can be evaluated using a calculator. Accordingly, this operation is central to converting from the logarithmic form back to the exponent form, helping us to find explicit values for the variable in question.

It is important to remember that the natural logarithm is the inverse of exponentiation by \( e \); therefore, solving equations with natural logarithms often includes using exponentiation to 'undo' the logarithm and isolate the variable.

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. For example, \( 3^2 \) means 3 is multiplied by itself twice, giving us 9.

When dealing with logarithms, exponentiation is the process used to 'undo' the logarithm. Since logarithmic functions are the inverse of exponential functions, you can solve a logarithmic equation by raising the base of the logarithm to the power given by the logarithm. This is why, in the exercise solution, exponentiation is used to isolate \( x \) on one side by applying \( e \) to the power of 1.6, denoted as \( e^{1.6} \).

In our exercise, we see that to find the value of \( x \), we must exponentiate the base \( e \) to the power of the given number. Calculator assists in finding the numerical value of such expressions, which in this case provides the approximate value of \( x \). It's essential for students to become comfortable with exponentiation as it is a fundamental operation that often appears in algebra and higher-level mathematics.