Variables \(y\) and \(t\) are thought to be connected by a law of the form $$ y=\frac{a^{t}}{k} $$ Measurements of \(t\) and \(y\) are (a) By using appropriate graph paper find the law connecting \(y\) and \(t\). (b) Predict \(y\) when \(t=12\). (c) Predict the value of \(t\) when \(y\) first exceeds 1000

Understanding logarithmic properties is essential when dealing with exponential equations like the one presented in the example exercise. A logarithm is the inverse operation to exponentiation and it answers the question: 'To what exponent must we raise a given number (the base) to obtain another number?'.

When we have equations of the form \( y = \frac{a^t}{k} \), taking the natural logarithm of both sides allows us to utilize several logarithmic properties to simplify and solve the equation. For instance, logarithms turn products into sums and quotients into differences, making complex exponential equations much more manageable. In the exercise, logarithmic properties were used to transform the equation into \( \ln(y) = t\ln(a) - \ln(k) \).

Some key logarithmic properties include:

• \( \log(ab) = \log(a) + \log(b) \)
• \( \log(\frac{a}{b}) = \log(a) - \log(b) \)
• \( \log(a^b) = b\log(a) \)
• For natural logarithms, \( e \) is the base, so \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \)

By using these properties, we can transform and solve for variables within logarithmic and exponential expressions with ease.

Plotting data points on a graph is a visual way of examining relationships between variables. In the exercise, measurements of \( t \) and \( y \) must be plotted on graph paper to establish a relationship between them. When plotting, each pair of \( t \) and \( y \) values forms a coordinate \( (t, y) \) that is placed on a grid.

Once all data points are plotted, patterns may emerge. In many cases, trend lines or curves can suggest the mathematical relationship between the variables. Properly plotted and analyzed graphs can reveal linear trends, cyclic patterns, or even the presence of outliers.

To facilitate the plotting of an exponential relationship on a linear graph, one can take the logarithm of the exponential variable, turning the exponential growth into a linear increase which is much easier to graph and interpret.

The line of best fit, also known as the trendline, is a straight line drawn through the data on a scatter plot that best expresses the relationship between the variables. It may not pass through all the points, but it aims to distribute them evenly on either side. The goal is to minimize the distance between the points and the line.

The formula for a line in slope-intercept form is \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) represents the y-intercept. By using the line of best fit, we can easily interpolate or extrapolate data, which in the context of the exercise means making predictions about \( y \) for a given \( t \), or vice versa.

In the given exercise, the line of best fit represents the logarithmic relationship between \( t \) and \( y \) after they have been linearized through logarithms. The slope of the line corresponds to \( \ln(a) \) and the y-intercept corresponds to \( -\ln(k) \). These values can then be used to determine the constants \( a \) and \( k \) in the original exponential equation, facilitating predictions and further analysis of the data.