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Stat 230 Notes

E Math review

E.1 Linear equations

E.2 Logarithms

Let \(b>0\) and \(x>0\). The logarithm (base-\(b\)) of \(x\) is denoted \(\log_b(x)\) and equal to \[ \log_b(x) = a \] where \(a\) tells us what power we must raise \(b\) to to obtain the value \(x\): \[ b^a = x \]

Easy examples are: \(b=2\), \(x=8\) and \(a=3\), \[ \log_2(8) = 3 \] since \(2^3 = 8\). Or using base \(b=10\), then \[ \log_{10}(0.01) = -2 \] since \(10^{-2} = 0.01\).

Some basic facts logarithm facts are \[ \log_b(b) = 1 \] since \(b^1 = b\) and \[ \log_b(1) = 0 \] since \(b^0 = 1\).

E.2.1 Interpreting logged variables

Multiplicative changes in \(x\) result in additive changes in \(\log_b(x)\). If \(m>0\), then \[ \log_b(mx) = \log_b(m) + \log_b(x) \] For example, a doubling of \(x=8\) results in an increase in \(\log_2(8)\) of one unit: \[ \log_2(16) = \log_2(2\times 8) = \log_2(2) + \log_2(8) = 1 + 3 = 4 \] More generally if we use a base-2 logarithm, a doubling of \(x\) results in an additive increase in \(\log(x)\) of 1 unit: \[ \log_2(2\times x) = \log_2(2) + \log_2(x) = 1 + \log_2(x) \]

E.2.2 Inverse (i.e. reversing the log, getting rid of the log, …)

The logarithm and exponential functions are inverses of one another. This means we can “get rid” of the log by calculating \(b\) raised to the logged-function: \[ b^{\log_b(x)} = x \] This will be useful in regression when we have a linear relationship between logged-response \(y\) and a set of predictors.

For example, suppose we know that \[ \log_2(y) = 3 + 5x + \epsilon \] To return this to an expression on the original (unlogged) scale of \(y\), we need take both sides raised to the base 2: \[ 2^{\log_2(y)} = 2^{3 + 5x+ \epsilon} \] Simplifying both sides gives \[ y = 2^3 \times 2^{5x} \times x^{\epsilon} \]

E.2.3 Logarithms in R

The R function log gives the natural logarithm (base-\(e\)):

log(2)
## [1] 0.6931472
log(exp(1))
## [1] 1

Other common logarithm bases are base-2 and base-10:

log2(8)
## [1] 3
log10(100)
## [1] 2

General bases can be added as an argument:

log(49, base = 7)
## [1] 2

E.3 Exercises

  1. Write the following as the sum of two logarithms. Simplify as much as possible:

    1. \(\log_2(2x)\)
    2. \(\log_2(0.5x)\)
    3. \(\ln(2x)\) where \(\ln\) is the natural log (base-\(e\))

  2. Write the following expressions in terms of \(y\), not \(\log(y)\). Simplify as much as possible:

    1. \(\log_2(y) = 1 - 3x\)
    2. \(\log_{10}(y) = -2 + 0.4x\)
    3. \(\ln(y) = 1 - 3x\)

  3. Write the following expressions in terms of \(y\) and \(x\), not \(\log(y)\) and \(\log(x)\). Simplify as much as possible:

    1. \(\log_2(y) = 1 - 3\log_2(x)\)
    2. \(\ln(y) = -2 + 0.4\ln(x)\)
    3. \(\ln(y) = 1 - 3\log_2(x)\)

  4. Logarithmic model: Regression of \(Y\) on \(\log(x)\) obtains the following estimated mean of \(Y\): \[ \hat{\mu}(Y \mid x) = 1 - 3 \log_2(x) \]

    1. What is the change in estimated mean response if we double the value of \(x\)?
    2. What is the change in estimated mean response if we triple the value of \(x\)?
    3. What is the change in estimated mean response if we reduce the value of \(x\) by 20%?

  5. Exponential model: Regression of \(\log_2(Y)\) on \(x\) obtains the following estimated median of \(Y\): \[ \hat{median}(\log_2(Y) \mid x) = -2 + 0.4x \]

    1. Write the median in terms of \(Y\) instead of \(\log_2(Y)\). Simplify as much as possible.
    2. What is the multiplicative change in estimated median response if we increase \(x\) by 1 unit?
    3. What is the percent change in estimated median response if we increase \(x\) by 1 unit?
    4. What is the multiplicative change in estimated median response if we decrease \(x\) by 2 units?
    5. What is the percent change in estimated median response if we decrease \(x\) by 2 units?

  6. Power model: Regression of \(\log_2(Y)\) on \(\log_2(x)\) obtains the following estimated median of \(Y\): \[ \hat{median}(\log_2(Y) \mid x) = 1 -3\log_2(x) \]

    1. Write the median in terms of \(Y\) and \(x\) instead of \(\log\)s. Simplify as much as possible.
    2. What is the multiplicative change in estimated median response if we increase \(x\) by 50%?
    3. What is the percent change in estimated median response if we increase \(x\) by 50%?
    4. What is the multiplicative change in estimated median response if we reduce the value of \(x\) by 20%?
    5. What is the percent change in estimated median response if we reduce the value of \(x\) by 20%?

Here is a link to the solution