P3 3.1, P3 3.2 - The exponential and natural logarithm

The function e to x and transformed graphs

Could $e^x$ ever equal zero or a negative number? It cannot: it is a positive-base exponential, so each of its outputs is positive, even when the exponent is negative.

Like the positive-base exponentials studied in P2, it obeys the index laws: $e^{u+v}=e^u e^v$ and $e^{-u}=1/e^u$. Since $e>1$, its graph increases: it tends to zero as $x$ becomes very negative and grows without bound as $x$ becomes very positive. The line $y=0$ is a horizontal asymptote, not part of the curve.

For $y=e^{ax+b}+c$ with $a\ne0$, the exponential part stays positive, so the domain is all real numbers, range $y>c$, and horizontal asymptote $y=c$. The curve increases if $a>0$ and decreases if $a<0$. Setting $ax+b=0$ gives a useful anchor with height $1+c$; setting $x=0$ gives the vertical-axis intercept $e^b+c$. If $a=0$, the expression is the constant $e^b+c$, a separate case.

[DIAGRAM: asset_name: Exponential graphs with a horizontal asymptote; asset_slug: exponential-transformations; caption: Changing the sign of x in the exponent reflects the curve in the vertical axis; both ranges remain above minus one.; description: Two accurately plotted curves y equals e to the power 2x minus 2 then minus one and y equals e to the power negative 2x minus 2 then minus one are reflections in the y axis, share intercept e to the power minus two minus one and horizontal asymptote y equals minus one, and cross the x axis at plus one and minus one respectively.]

Change only the sign of the coefficient of $x$ inside the exponent. This reflects the whole previous curve in the vertical axis.

A shift inside the exponent belongs to the exponent: $e^{2x-2}=e^{-2}e^{2x}$, whereas $e^{2x}-2$ is a different function.

Natural logarithm as inverse

The equation $e^x=5$ has one real solution because $e^x$ increases through every positive output. The natural logarithm, written $\ln$, names that exponent: $x=\ln5$.

Reflecting $y=e^x$ in $y=x$ gives $y=\ln x$. The points $(0,1),(1,e),(-1,1/e)$ reflect to $(1,0),(e,1),(1/e,-1)$. Hence logarithm inputs must be positive, but outputs may be positive, zero or negative. Its domain is $x>0$, range all real numbers, and vertical asymptote $x=0$. It increases, dropping without bound as a positive input approaches zero.

[DIAGRAM: asset_name: Exponential and logarithm as inverse graphs; asset_slug: exponential-log-inverse; caption: Swapping the coordinates reflects the exponential graph into the natural-log graph.; description: Equal-scale axes show y equals e to x and y equals natural log x reflected in y equals x. Corresponding points zero one and one zero, one e and e one are marked. The positive domain of natural logarithm and the exponential positive range are visible.]

Natural logarithms obey the P2 log laws, with the same positive-input conditions:

In particular, $\ln1=0$, $\ln e=1$, and $\ln(e^r)=r$. A sum inside a logarithm does not split: $\ln(a+b)$ is generally not $\ln a+\ln b$. The booklet supplies $e^{x\ln a}=a^x$ for positive $a$; it connects another positive base to the natural exponential.