Chapter FourDescribing the Relation Between Two Variables

Section 4.4

Nonlinear Regression: Transformations

EXAMPLE Using the Definition of a Logarithm

Rewrite the logarithmic expressions to an equivalent expression involving an exponent. Rewrite the exponential expressions to an equivalent logarithmic expression.

(a) log315 = a (b) 45 = z

In the following properties, M, N, and a are positive real numbers, with a 1, and r is any real number.

loga (MN) = loga M + loga N

loga Mr = r loga M

EXAMPLE Simplifying Logarithms

Write the following logarithms as the sum of logarithms. Express exponents as factors.

(a) log2 x4 (b) log5(a4b)

If a = 10 in the expression y = logax, the resulting logarithm, y = log10x is called the common logarithm. It is common practice to omit the base, a, when it is equal to 10 and write the common logarithm as y = log x

EXAMPLE Evaluating Exponential and Logarithmic Expressions

Evaluate the following expressions. Round your answers to three decimal places.

(a) log 23 (b) 102.6

y = abx Exponential Model

log y = log (abx) Take the common logarithm of both sides

log y = log a + log bx

log y = log a + x log b

Y = A + B x where

b = 10B a = 10A

EXAMPLE 4 Finding the Curve of Best Fit to an Exponential Model

A chemist as a 1000-gram sample of a radioactive material. She records the amount of radioactive material remaining in the sample every day for a week and obtains the following data.

DayDay Weight (in grams)Weight (in grams)0 1000.01 897.12 802.53 719.84 651.15 583.46 521.77 468.3

(a) Draw a scatter diagram of the data treating the day, x, as the predictor variable.

(b) Determine Y = log y and draw a scatter diagram treating the day, x, as the predictor variable and Y = log y as the response variable. Comment on the shape of the scatter diagram.

(c) Find the least-squares regression line of the transformed data.

(d) Determine the exponential equation of best fit and graph it on the scatter diagram obtained in part (a).

(e) Use the exponential equation of best fit to predict the amount of radioactive material is left after 8 days.

y = axb Power Model

log y = log (axb) Take the common logarithm of both sides

log y = log a + log xb

log y = log a + b log x

Y = A + b X where a = 10A

EXAMPLE Finding the Curve of Best Fit to a Power Model

Cathy wishes to measure the relation between a light bulb’s intensity and the distance from some light source. She measures a 40-watt light bulb’s intensity 1 meter from the bulb and at 0.1-meter intervals up to 2 meters from the bulb and obtains the following data.

DistanceDistance IntensityIntensity1.0 0.09721.1 0.08041.2 0.06741.3 0.05721.4 0.04951.5 0.04331.6 0.03841.7 0.03391.8 0.02941.9 0.02682.0 0.0224

(a) Draw a scatter diagram of the data treating the distance, x, as the predictor variable.

(b) Determine X = log x and Y = log y and draw a scatter diagram treating the day, X = log x, as the predictor variable and Y = log y as the response variable. Comment on the shape of the scatter diagram.

(c) Find the least-squares regression line of the transformed data.

(d) Determine the power equation of best fit and graph it on the scatter diagram obtained in part (a).

(e) Use the power equation of best fit to predict the intensity of the light if you stand 2.3 meters away from the bulb.

Modeling is not only a science but also an art form. Selecting an appropriate model requires experience and skill in the field in which you are modeling. For example, knowledge of economics is imperative when trying to determine a model to predict unemployment. The main reason for this is that there are theories in the field that can help the modeler to select appropriate relations and variables.