Indicates an exponential relationship between the x and y variables.

A faster rate of exponential growth or decay.

Indicates that the y-value is constant (no growth or decay).

Find the y-intercept of the linearized data and calculate its antilog.

Exponential growth.

Exponential decay.

Different bases will change the scale of the y-axis, affecting the slope's numerical value but not the overall trend.

The relationship is not purely exponential; it may be polynomial, logarithmic, or a combination of functions.

Compare the slopes of the lines representing each dataset. The steeper the slope, the faster the growth rate.

The data may have some error or noise, or the relationship may not be perfectly exponential.

$$\log_n(y) = \log_n(a) + x \log_n(b)$$\n$$y = (\log_n b)x + \log_n(a)$$

$$y = mx + b$$

$$m = \frac{\log(y_2) - \log(y_1)}{x_2 - x_1}$$

$$a = n^b$$, where n is the base of the logarithm.

$$y = 10^{mx+b}$$ (if base 10 logarithm)

$$y = ae^{kt}$$, where a is the initial value, k is the growth rate, and t is time.

$$y = ae^{-kt}$$, where a is the initial value, k is the decay rate, and t is time.

The slope and y-intercept of the linearized equation can be used to determine the parameters of the original exponential equation.

$$\log(ab) = \log(a) + \log(b)$$

$$\log(a^x) = x\log(a)$$

Use a semi-log plot when the data spans several orders of magnitude or when you suspect exponential growth or decay.

Exponential relationships appear as straight lines on a semi-log plot, making them easier to identify.

Because the logarithmic scale compresses the y-values, making the exponential relationship linear.

Changing the base affects the slope and y-intercept of the linearized data but doesn't change the linearity.

The slope represents the rate of exponential growth or decay. A positive slope indicates growth, and a negative slope indicates decay.

The y-intercept represents the logarithm of the initial value of the exponential function.

Biology (bacterial growth), chemistry (reaction kinetics), physics (radioactive decay), and finance (compound interest).

The linear x-axis preserves the proportionality of the independent variable, which is essential for interpreting the rate of change.

By transforming exponential relationships into linear ones, it simplifies the process of finding rates of change and initial values.

It is only suitable for data that exhibits exponential behavior. It may not be useful for other types of relationships.