The Logarithm Laws | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

Multiplying results adds their exponents

If $u=a^p$ and $v=a^q$ , multiplying gives $uv=a^{p+q}$ . A logarithm asks for that exponent, so

\log_a(uv)=\log_a u+\log_a v

For example, $2^3\times2^2=2^5$ becomes $\log_2(8\times4)=3+2=5$ .

Dividing powers subtracts exponents; raising a power to another power multiplies them. These give the other two laws.

\begin{gathered}\frac{a^p}{a^q}=a^{p-q}\\\log_a\left(\frac uv\right)=\log_a u-\log_a v\end{gathered}

\begin{gathered}(a^p)^t=a^{pt}\\\log_a(u^t)=t\log_a u\end{gathered}

Here $a>0$ , $a\ne1$ , and $u,v>0$ . Use the same base throughout. All three laws also work for $\ln$ .

Your turnPractice

Without a calculator, simplify $\log_2 3+\log_2 8-\log_2 6$ and $\ln(e^4)$ .

Show worked answer

\log_2\left(\frac{3\times8}{6}\right)=\log_2 4=2

Answer

2\quad\text{and}\quad4

There is no law for splitting a sum

A product inside a log becomes a sum of logs. A sum inside a log does not split.

\begin{gathered}\ln(1+1)=\ln2\ne0\\\ln1+\ln1=0\end{gathered}

Likewise, $(\ln x)^2$ means square the logarithm’s value. It is different from $\ln(x^2)$ , which means take the log of the squared input.

Your turnPractice

At $x=e^3$ , evaluate $(\ln x)^2$ and $\ln(x^2)$ . Are they equal?

Show worked answer

Answer

9\quad\text{and}\quad6

$(\ln(e^3))^2=3^2$ , whereas $\ln(e^6)=6$ .

Multiplying results adds their exponents

If $u=a^p$ and $v=a^q$ , multiplying gives $uv=a^{p+q}$ . A logarithm asks for that exponent, so

\log_a(uv)=\log_a u+\log_a v

For example, $2^3\times2^2=2^5$ becomes $\log_2(8\times4)=3+2=5$ .

Dividing powers subtracts exponents; raising a power to another power multiplies them. These give the other two laws.

\begin{gathered}\frac{a^p}{a^q}=a^{p-q}\\\log_a\left(\frac uv\right)=\log_a u-\log_a v\end{gathered}

\begin{gathered}(a^p)^t=a^{pt}\\\log_a(u^t)=t\log_a u\end{gathered}

Here $a>0$ , $a\ne1$ , and $u,v>0$ . Use the same base throughout. All three laws also work for $\ln$ .

Your turnPractice

Without a calculator, simplify $\log_2 3+\log_2 8-\log_2 6$ and $\ln(e^4)$ .

Show worked answer

\log_2\left(\frac{3\times8}{6}\right)=\log_2 4=2

Answer

2\quad\text{and}\quad4

There is no law for splitting a sum

A product inside a log becomes a sum of logs. A sum inside a log does not split.

\begin{gathered}\ln(1+1)=\ln2\ne0\\\ln1+\ln1=0\end{gathered}

Likewise, $(\ln x)^2$ means square the logarithm’s value. It is different from $\ln(x^2)$ , which means take the log of the squared input.

Your turnPractice

At $x=e^3$ , evaluate $(\ln x)^2$ and $\ln(x^2)$ . Are they equal?

Show worked answer

Answer

9\quad\text{and}\quad6

$(\ln(e^3))^2=3^2$ , whereas $\ln(e^6)=6$ .