A graph with one logarithmic axis and one linear axis.

The y-axis represents the logarithm of the data values.

Exponential data.

To visualize data with a wide range of values and identify exponential trends.

Transforming exponential data into a linear form by taking the logarithm of the dependent variable.

The rate of growth or decay of the exponential function.

The logarithm of the initial value of the exponential function.

A linear scale has equal intervals, while a logarithmic scale compresses larger values.

Base 10 or base e (natural logarithm).

The x-axis is always on a linear scale.

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

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

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

1. Plot the independent variable on the linear x-axis. 2. Take the logarithm of the dependent variable. 3. Plot the logarithm values on the logarithmic y-axis.

1. Take the logarithm of the y-values. 2. Plot the x-values against the log(y)-values. 3. Fit a straight line to the plotted points.

1. Calculate the slope (m) using two points on the line. 2. Determine the y-intercept (b). 3. Write the equation in the form log(y) = mx + b.

1. Find the y-intercept (b) of the linearized plot. 2. Calculate the antilog of the y-intercept (10^b or e^b depending on the log base).

1. Calculate the slope (m) of the linearized plot. 2. The decay rate is the absolute value of the slope.

1. Linearize the data and find the equation of the line. 2. Plug in the future x-value into the equation. 3. Calculate the antilog of the resulting y-value to find the predicted value.

1. Examine the data for exponential trends. 2. Check if the y-values span several orders of magnitude. 3. If both conditions are met, a semi-log plot is appropriate.

1. Identify the slope (m) and y-intercept (b) of the linearized equation. 2. The original equation is y = (10^b)(10^m)^x (if base 10 logarithm).

Semi-log plots are not suitable for negative y-values, as the logarithm of a negative number is undefined. You may need to transform the data or use a different type of plot.

Plot both functions on the same semi-log plot. Compare their slopes and y-intercepts to analyze their relative growth or decay rates and initial values.