Natural Logarithms and Calculator Keys | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

ln is a logarithm to base e

$e$ is a fixed number, approximately 2.71828, not a variable. The natural logarithm uses this base: $\ln x=\log_e x$ . The function $e^x$ is the natural exponential.

e^p=q\quad\Longleftrightarrow\quad\ln q=p

They undo each other, so they are inverse functions. Logarithms return the exponent; exponentials restore the original positive input.

\begin{gathered}\ln(e^x)=x\quad\text{for every real }x\\e^{\ln x}=x\quad\text{for }x>0\end{gathered}

Worked example

Simplify without a calculator.

\begin{gathered}\ln(e^{-3})=-3\\e^{\ln 7}=7\\e^{2\ln 5}=(e^{\ln5})^2=25\end{gathered}

The last step uses the index rule $(a^u)^v=a^{uv}$ : a power raised to another power multiplies its exponents.

Your turnPractice

Simplify $\ln 1$ , $\ln e$ and $e^{-\ln4}$ without a calculator.

Show worked answer

Answer

0,\quad1,\quad\frac14

Use $e^0=1$ , $e^1=e$ and $e^{-\ln4}=1/e^{\ln4}$ .

Use the matching calculator key

On a calculator, log means base 10 and ln means base $e$ . The $10^x$ and $e^x$ functions undo them; these may be shifted keys or menu functions on your model.

Worked example

Solve 10ˣ = 6 and ln x = −2.

\begin{gathered}10^x=6\\x=\log_{10}6\approx0.778\\[4pt]\ln x=-2\\x=e^{-2}\approx0.135\end{gathered}

These decimals are to 3 significant figures: count from the first non-zero digit. The logarithm and exponential expressions are exact; the decimals are approximations.

Enter the whole log input or exponent in brackets. Keep the full calculator value until the last line. For an exact answer, leave $\ln6$ or $e^{-2}$ as it is.

Your turnPractice

Solve $\ln x=1.4$ and $10^y=0.2$ . Give exact answers, then 3 significant figures.

Show worked answer

\begin{gathered}x=e^{1.4}\approx4.06\\y=\log_{10}0.2\approx-0.699\end{gathered}

ln is a logarithm to base e

$e$ is a fixed number, approximately 2.71828, not a variable. The natural logarithm uses this base: $\ln x=\log_e x$ . The function $e^x$ is the natural exponential.

e^p=q\quad\Longleftrightarrow\quad\ln q=p

They undo each other, so they are inverse functions. Logarithms return the exponent; exponentials restore the original positive input.

\begin{gathered}\ln(e^x)=x\quad\text{for every real }x\\e^{\ln x}=x\quad\text{for }x>0\end{gathered}

Worked example

Simplify without a calculator.

\begin{gathered}\ln(e^{-3})=-3\\e^{\ln 7}=7\\e^{2\ln 5}=(e^{\ln5})^2=25\end{gathered}

The last step uses the index rule $(a^u)^v=a^{uv}$ : a power raised to another power multiplies its exponents.

Your turnPractice

Simplify $\ln 1$ , $\ln e$ and $e^{-\ln4}$ without a calculator.

Show worked answer

Answer

0,\quad1,\quad\frac14

Use $e^0=1$ , $e^1=e$ and $e^{-\ln4}=1/e^{\ln4}$ .

Use the matching calculator key

On a calculator, log means base 10 and ln means base $e$ . The $10^x$ and $e^x$ functions undo them; these may be shifted keys or menu functions on your model.

Worked example

Solve 10ˣ = 6 and ln x = −2.

\begin{gathered}10^x=6\\x=\log_{10}6\approx0.778\\[4pt]\ln x=-2\\x=e^{-2}\approx0.135\end{gathered}

These decimals are to 3 significant figures: count from the first non-zero digit. The logarithm and exponential expressions are exact; the decimals are approximations.

Enter the whole log input or exponent in brackets. Keep the full calculator value until the last line. For an exact answer, leave $\ln6$ or $e^{-2}$ as it is.

Your turnPractice

Solve $\ln x=1.4$ and $10^y=0.2$ . Give exact answers, then 3 significant figures.

Show worked answer

\begin{gathered}x=e^{1.4}\approx4.06\\y=\log_{10}0.2\approx-0.699\end{gathered}