3.2 Logarithmic and exponential functions

Learning objectives

3.2.1Log laws• understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)3.2.2Exponential and ln• understand the definition and properties of ex and ln x, including their relationship as inverse functions and their graphs Including knowledge of the graph of y = ekx for both positive and negative values of k.3.2.3Log equations• use logarithms to solve equations and inequalities in which the unknown appears in indices e.g. 25x 1, 32 5<x31# -, 34xx 12 1= -+.3.2.4Linearising relationships• use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. e.g. y = kxn gives ln y = ln k + n ln x which is linear in ln x and ln y. y = k (ax) gives ln y = ln k + x ln a which is linear in x and ln y.

$For $a>0$, $a e1$ and $y>0$:$ a^x=y\quad\Longleftrightarrow\quad \log_a y=x. $Hence $\log_a1=0$ and $\log_a a=1$.$

Structure	Logarithm law (positive arguments)
product	$\log_a(MN)=\log_aM+\log_aN$
quotient	$\log_a(M/N)=\log_aM-\log_aN$
power	$\log_a(M^p)=p\log_aM$

Record the positivity conditions before combining or expanding logs. Move coefficients into powers when useful, combine to one logarithm, then translate back to index form.

$For $x>0$,$ \log_2(8x)-\log_2x=\log_2(8)=3.

$\log(a+b)$ does not split. Change-of-base formulae are explicitly excluded from this objective.

y=e^x\quad\Longleftrightarrow\quad x=\ln y. $Thus $\ln(e^x)=x$ for real $x$, while $e^{\ln x}=x$ requires $x>0$. Their graphs reflect in $y=x$.$

Graph	Domain	Range	Intercept/asymptote
$y=e^{kx}$ , $k
e0 $\| all real$ x $\|$ y>0 $\|$ (0,1) $; horizontal asymptote$ y=0$
$y=\ln x$	$x>0$	all real $y$	$(1,0)$ ; vertical asymptote $x=0$

For $k>0$ , $e^{kx}$ increases; for $k<0$ , it decreases. Both stay positive and approach, but never cross, the $x$ -axis in one direction.

$$3e^{2x}=12$ gives $e^{2x}=4$, so $x= frac12\ln4$.$

An exponential graph has no $x$ -intercept, and $\ln x$ is undefined for $x\le0$ . Do not treat $\ln(x^2)=2\ln x$ as valid when $x<0$ .

Rearrange until each exponential expression is positive, take $\ln$ of both sides, use $\ln(a^{g(x)})=g(x)\ln a$ , then solve the resulting algebraic equation and check in the original.

5^{2x-1}=12\quad\Rightarrow\quad(2x-1)\ln5=\ln12\quad\Rightarrow\quad x=rac{1+\ln12/\ln5}{2}.

Base form	Monotonic direction
$a^{u}<a^{v}$ with $a>1$	$u<v$
$a^{u}<a^{v}$ with $0<a<1$	$u>v$

If terms such as $a^{2x}$ and $a^x$ occur together, set $t=a^x$ with $t>0$ , solve the polynomial in $t$ , reject non-positive roots, then take logs to recover $x$ .

Logs can only be taken after both sides are positive. An inequality reverses for a decreasing base $0<a<1$ , not merely because logarithms were used.

Transform variables so the model becomes Y=mX+c. For y=ab^x, plotting ln y against x gives gradient ln b and intercept ln a; for y=ax^n, plotting ln y against ln x gives gradient n.

Transform measured uncertainties and units consistently, then interpret the gradient and intercept in the original parameters. A straight plot supports the model but does not prove causation.

If ln y=0.7x+1.2, then a=e^{1.2} and b=e^{0.7} for y=ab^x.

The intercept is not always the original constant; it may be ln a or another transformed quantity.

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—3.3 TrigonometryRead

3.2 Logarithmic and exponential functions

Learning objectives

Translate index statements with the three logarithm laws

Read exponential and natural-log graphs as inverse shapes

Bring an unknown exponent down with logarithms

Linearising a relationship makes a model testable with a straight-line graph

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