If a relationship can be written in the form \[ Y = mX + c, \] then it is linear and can be represented by a straight line.

We use logarithms to turn some curved relationships into this form.

These laws help us rewrite expressions into the form \(Y=mX+c\).

Form 1: \(y=ax^n\)

Take logarithms: \[ \log y=\log(ax^n) \] \[ \log y=\log a+\log(x^n) \] \[ \log y=n\log x+\log a \]

So if we plot \( \log y \) against \( \log x \), we get a straight line with:

• gradient \(= n\)
• intercept \(= \log a\)

Form 2: \(y=ab^x\)

Take logarithms: \[ \log y=\log(ab^x) \] \[ \log y=\log a+x\log b \]

So if we plot \( \log y \) against \( x \), we get a straight line with:

• gradient \(= \log b\)
• intercept \(= \log a\)

Show that \(y=5x^3\) can be written in linear form.

Take logarithms: \[ \log y=\log(5x^3) \] \[ \log y=\log 5+\log x^3 \] \[ \log y=3\log x+\log 5 \]

Show that \(y=2\cdot 4^x\) can be written in linear form.

Take logarithms: \[ \log y=\log(2\cdot 4^x) \] \[ \log y=\log 2+\log 4^x \] \[ \log y=x\log 4+\log 2 \]