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Logarithmic TransformationsBasic Properties of Logs: Pg. 772

Product logb mn = logb m + logb n

Quotient logb = logb m – logb n

Power logb mp = p logb m

Equality if logb m = logb n, then m = n

Just like multiplication is the inverse of division and squares are the inverse of square roots, logs have inverses too.

The inverse of the most common log is 10x (base 10) 10x

cancels log The inverse of a natural log ln is ex (base e) e cancels ln

Logarithmic TransformationsTo solve logs, you can use the log and ln buttons on your calculator. But when x is in a log, you must rewrite it or “undo” or invert it.

You can rewrite the logs in exponential form:logb x = y → by = x

log x = 2 102 = x = 100 (when there is no base, the log is in base 10)

ln x = y → ey = xln x = 3 e3 = x

Logarithmic TransformationsWhen you use logs to transform data to make a linear graph, you have two options:1. Use the log or natural log of the y variable (response

variable) – log transformation & exponential models. The graph will be the explanatory variable x on the x-axis, and the natural log of the response variable ln y or log y on the y-axis.

2. Use the log of both variables – power models using log transformations. The x-axis is the log or natural log of the explanatory variable log x or ln x, and the y-axis is the log or natural log of the response variable log y or ln y.

Logarithmic TransformationsReading & Interpreting option 1 - Log transformations on the response variable y.

Pg. 773-775

If a variable grows exponentially, its logarithm grows linearly.

Logarithmic TransformationsOption 2 - Power Models with 2 log transformations

If a power model describes the relationship between two variables, a scatterplot of the logarithms of both variables should produce a linear pattern.

Pg. 778-779

Logarithmic TransformationsWe can use our graphing calculators to model log transformations and determine whether a log transformation is needed on just y or both x and y to achieve linearity.

Enter the values of the explanatory data in L1 and the values of the response variable in L2

Define L3 as the log of L1Define L4 as the log of L2

To determine which model to use, graph aplot of L1 vs L4, then graph a plot of L3 vs L4

We will use whichever is more linear.

Run Linear Regression – LinReg(a + bx) L3, L4, Y1

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