Combining and Expanding Logarithms | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

Move coefficients inside before combining

To write several logs as one log, first turn each outside coefficient into a power of its input. Then use products for addition and quotients for subtraction.

Worked example

Write 2 ln x − ½ ln y as a single logarithm.

With $x,y>0$ , the half power is a square root:

\begin{gathered}2\ln x-\tfrac12\ln y\\=\ln(x^2)-\ln(\sqrt y)\\=\ln\left(\frac{x^2}{\sqrt y}\right)\end{gathered}

A number without a log can also be written as a log: $1=\ln e$ , or $2=\log_3 9$ . Match the base already used.

Your turnPractice

Write $1+2\log_2 x-\log_2 y$ as one log, for $x,y>0$ .

Show worked answer

\log_2 2+\log_2 x^2-\log_2 y

Answer

\log_2\left(\frac{2x^2}{y}\right)

Expand only products, quotients and powers

Worked example

Expand ln(3x² / √(x + 1)), for x > 0.

\ln3+2\ln x-\tfrac12\ln(x+1)

The $x+1$ stays inside its own log: it is a sum, not a product.

Keep the original restrictions when you combine logs. Although $\ln(x^2)$ is defined for negative $x$ too, $2\ln x$ is only defined for $x>0$ . They are equal on that original domain, not for every non-zero $x$ .

Your turnPractice

Given $\log_4 p=u$ and $\log_4 q=v$ , express $\log_4(64\sqrt p/q^2)$ in terms of $u,v$ .

Show worked answer

\log_4 64+\tfrac12\log_4p-2\log_4q

Answer

3+\tfrac12u-2v

Both $p$ and $q$ are positive because their original logs exist.

Move coefficients inside before combining

To write several logs as one log, first turn each outside coefficient into a power of its input. Then use products for addition and quotients for subtraction.

Worked example

Write 2 ln x − ½ ln y as a single logarithm.

With $x,y>0$ , the half power is a square root:

\begin{gathered}2\ln x-\tfrac12\ln y\\=\ln(x^2)-\ln(\sqrt y)\\=\ln\left(\frac{x^2}{\sqrt y}\right)\end{gathered}

A number without a log can also be written as a log: $1=\ln e$ , or $2=\log_3 9$ . Match the base already used.

Your turnPractice

Write $1+2\log_2 x-\log_2 y$ as one log, for $x,y>0$ .

Show worked answer

\log_2 2+\log_2 x^2-\log_2 y

Answer

\log_2\left(\frac{2x^2}{y}\right)

Expand only products, quotients and powers

Worked example

Expand ln(3x² / √(x + 1)), for x > 0.

\ln3+2\ln x-\tfrac12\ln(x+1)

The $x+1$ stays inside its own log: it is a sum, not a product.

Your turnPractice

Given $\log_4 p=u$ and $\log_4 q=v$ , express $\log_4(64\sqrt p/q^2)$ in terms of $u,v$ .

Show worked answer

\log_4 64+\tfrac12\log_4p-2\log_4q

Answer

3+\tfrac12u-2v

Both $p$ and $q$ are positive because their original logs exist.