Simultaneous Logarithmic Equations | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Name both logarithms to get linear equations

Simultaneous equations must hold for the same pair of values. Repeated logarithms such as $\ln x$ and $\ln y$ can be given short names.

Worked example

Solve ln(x²y) = 4 and ln(x/y) = −1, for x,y > 0.

Write $u=\ln x$ and $v=\ln y$ before expanding each log with the product, quotient and power laws.

\begin{gathered}2u+v=4,\\u-v=-1.\end{gathered}

Add the equations: the $v$ terms cancel, giving $3u=3$ . Then $u=1$ and $v=2$ .

Answer

x=e,\qquad y=e^2

Check both: $x^2y=e^4$ and $x/y=e^{-1}$ .

Your turnPractice

For positive $x,y$ , solve $\ln(xy)=3$ and $\ln(x/y)=1$ .

Show worked answer

\begin{gathered}u+v=3,\quad u-v=1\\2u=4\\u=2,\quad v=1\end{gathered}

Answer

x=e^2,\qquad y=e

Convert a log equation before substituting

Worked example

Solve xy = 81 and logₓ y = 3.

The log requires $x>0$ , $x\ne1$ and $y>0$ . Rewrite it as $y=x^3$ and substitute into the product equation.

x(x^3)=81\Rightarrow x^4=81

The real fourth roots are 3 and −3, but only the positive base is allowed.

Answer

x=3,\qquad y=27

$3\times27=81$ and $3^3=27$ , so both equations hold.

Your turnPractice

Solve $xy=16$ and $\log_x y=3$ in real numbers.

Show worked answer

y=x^3,\quad x^4=16,\quad x>0

Answer

x=2,\qquad y=8

Name both logarithms to get linear equations

Simultaneous equations must hold for the same pair of values. Repeated logarithms such as $\ln x$ and $\ln y$ can be given short names.

Worked example

Solve ln(x²y) = 4 and ln(x/y) = −1, for x,y > 0.

Write $u=\ln x$ and $v=\ln y$ before expanding each log with the product, quotient and power laws.

\begin{gathered}2u+v=4,\\u-v=-1.\end{gathered}

Add the equations: the $v$ terms cancel, giving $3u=3$ . Then $u=1$ and $v=2$ .

Answer

x=e,\qquad y=e^2

Check both: $x^2y=e^4$ and $x/y=e^{-1}$ .

Your turnPractice

For positive $x,y$ , solve $\ln(xy)=3$ and $\ln(x/y)=1$ .

Show worked answer

\begin{gathered}u+v=3,\quad u-v=1\\2u=4\\u=2,\quad v=1\end{gathered}

Answer

x=e^2,\qquad y=e

Convert a log equation before substituting

Worked example

Solve xy = 81 and logₓ y = 3.

The log requires $x>0$ , $x\ne1$ and $y>0$ . Rewrite it as $y=x^3$ and substitute into the product equation.

x(x^3)=81\Rightarrow x^4=81

The real fourth roots are 3 and −3, but only the positive base is allowed.

Answer

x=3,\qquad y=27

$3\times27=81$ and $3^3=27$ , so both equations hold.

Your turnPractice

Solve $xy=16$ and $\log_x y=3$ in real numbers.

Show worked answer

y=x^3,\quad x^4=16,\quad x>0

Answer

x=2,\qquad y=8