IB AA: Exponentials and Logarithms

Exponential and logarithmic functions are among the most powerful mathematical tools you will encounter, describing everything from viral spread and compound interest to seismic activity and sound intensity. Mastering their laws, forms, and applications is essential not only for the IB AA exam but also for interpreting the quantitatively-driven world around you.

Foundations: The Laws of Exponents and Logarithms

All work with exponentials and logarithms rests on a firm set of algebraic laws. These are not arbitrary rules but logical consequences of how these operations are defined, and fluency with them is non-negotiable.

Exponent Laws govern expressions where a number, the base, is raised to a power. For a base $a>0$ and real numbers $m$ and $n$, the key laws are:

• Product of Powers: $a^m\times a^n=a^{m+n}$
• Quotient of Powers: $a^m÷a^n=a^{m-n}$
• Power of a Power: $(a^m{)}^n=a^{m\times n}$
• Power of a Product: $(ab{)}^n=a^n{b}^n$
• Zero and Negative Exponents: $a^0=1$ and $a^{-n}=\frac{1}{a^n}$

Logarithm Laws are the direct counterparts to exponent laws because a logarithm answers the question: "To what exponent must we raise the base to get a certain number?" By definition, if $a^c=b$, then ${\log }_{a}b=c$, where $a$ (the base) is positive and not equal to 1. The principal laws are:

• Product Law: ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$
• Quotient Law: ${\log }_a(\frac{x}{y})={\log }_{a}x-{\log }_{a}y$
• Power Law: ${\log }_a(x^n)=n{\log }_{a}x$
• Change of Base: ${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$. This incredibly useful formula allows you to evaluate logarithms of any base using a calculator, which typically only has keys for ${\log }_{10}$ (log) and ${\log }_e$ (ln).

A critical insight is that logarithms and exponents are inverse functions. This means ${\log }_a(a^x)=x$ and $a^{{\log }_{a}x}=x$ for $x>0$. This inverse relationship is the key to solving equations.

Solving Exponential and Logarithmic Equations

The strategy for solving these equations hinges on using the laws to express both sides of the equation in a comparable form, then leveraging the inverse relationship.

For exponential equations (where the variable is in the exponent), like $2^{x+1}=5$, you often take the logarithm of both sides. This allows you to "bring down" the exponent. $$\begin{array}{rl}{2}^{x+1}& =5\\ \log (2^{x+1})& =\log (5)\text{(Applying log base 10 to both sides)}\\ (x+1)\log (2)& =\log (5)\text{(Using the Power Law)}\\ x+1& =\frac{\log (5)}{\log (2)}\\ x& =\frac{\log (5)}{\log (2)}-1\end{array}$$

For logarithmic equations (like ${\log }_3(x-4)+{\log }_3(x)=1$), you first use the logarithm laws to condense into a single log expression, then rewrite in exponential form to solve. $$\begin{array}{rl}{\log }_3((x-4)(x))& =1\text{(Product Law)}\\ (x-4)(x)& =3^1\text{(Rewriting in exponential form)}\\ x^2-4x& =3\\ x^2-4x-3& =0\end{array}$$ You then solve the quadratic, always checking your solutions in the original equation, as the domain of ${\log }_3(x-4)$ and ${\log }_3(x)$ requires $x>4$. Any solution not satisfying this is extraneous.

The Natural Logarithm and Exponential Growth Models

A special base, $e$ (Euler's number, approximately 2.71828), arises naturally in calculus and continuous growth processes. The function $e^x$ is the natural exponential function, and its inverse, ${\log }_{e}x$, is the natural logarithm, written $\ln x$.

This leads to the most important continuous growth and decay model: $N(t)=N_0{e}^{kt}$.

• $N(t)$ is the quantity at time $t$.
• $N_0$ is the initial quantity (when $t=0$).
• $k$ is the continuous growth/decay rate. If $k>0$, it's growth; if $k<0$, it's decay.
• The doubling time (for growth) or half-life (for decay) is constant and calculated as $T=\frac{\ln (2)}{|k|}$.

For example, a bacteria culture starting with 200 cells and growing at a continuous rate of 5% per hour is modeled by $N(t)=200e^{0.05t}$. To find when the population reaches 1000, you solve: $$\begin{array}{rl}200e^{0.05t}& =1000\\ e^{0.05t}& =5\\ 0.05t& =\ln (5)\text{(Taking the natural log of both sides)}\\ t& =\frac{\ln (5)}{0.05}\end{array}$$ This model contrasts with discrete compound growth ($A=P(1+\frac{r}{n}{)}^{nt}$), but for continuous compounding, the exponential model with base $e$ is used.

Graphical Analysis and Transformations

Understanding the shapes of these graphs solidifies your comprehension. The graph of $y=a^x$ ($a>1$) shows:

• Exponential growth: Increasing, passing through $(0,1)$, with a horizontal asymptote at $y=0$ as $x\to -\infty$.
• Exponential decay ($0<a<1$) is the mirror image: decreasing, with an asymptote at $y=0$ as $x\to \infty$.

The graph of $y={\log }_{a}x$ ($a>1$) shows:

• The inverse shape: Increasing, passing through $(1,0)$, with a vertical asymptote at $x=0$.

You must be comfortable with graphical transformations. The function $y=p\times a^{q(x+r)}+s$ represents:

• Vertical stretch/reflection by factor $p$
• Horizontal stretch/compression by factor $1/q$ and possible reflection
• Horizontal shift by $-r$
• Vertical shift by $s$ (this changes the horizontal asymptote from $y=0$ to $y=s$)

Similarly, for $y=p{\log }_a(q(x+r))+s$, the vertical asymptote shifts from $x=0$ to $x=-r$.

Common Pitfalls

1. Misapplying Logarithm Laws: The logarithm of a sum is not the sum of the logs. ${\log }_a(x+y)\ne {\log }_{a}x+{\log }_{a}y$. You can only separate logs for products and quotients using the Product and Quotient Laws.

• Correction: Condense expressions first. To solve ${\log }_2(x)+{\log }_2(3)=4$, correctly combine: ${\log }_2(3x)=4$, then $3x=2^4$.

1. Ignoring Domains: Logarithms are only defined for positive arguments, and the base must be positive and not equal to 1. When solving equations, any solution that results in taking the log of a non-positive number is invalid.

• Correction: Always state the domain restrictions at the outset and verify all potential solutions against them.

1. Confusing Exponential and Logarithmic Forms: Students sometimes mistakenly try to convert ${\log }_{a}b=c$ to $b^c=a$. This inverts the relationship.

• Correction: Remember the direct conversion: ${\log }_{a}b=c⟺a^c=b$. The base of the log becomes the base of the exponential.

1. Incorrectly Handling the Growth Rate k: In the model $N=N_0{e}^{kt}$, if a problem states "decays at 3% per year," then $k=-0.03$, not $0.03$.

• Correction: Read carefully. Growth implies a positive $k$, decay a negative $k$.

Summary

• Exponent and logarithm laws are inverse operations and provide the essential toolkit for manipulating related expressions and equations.
• The change of base formula (${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$) is indispensable for calculation and solving.
• To solve exponential equations, take the log of both sides to utilize the Power Law. To solve logarithmic equations, condense logs and convert to exponential form, always checking the domain.
• The natural logarithm ($\ln$) with base $e$ is central to the continuous growth/decay model $N(t)=N_0{e}^{kt}$, where $k$'s sign determines growth or decay.
• Graphically, exponential functions have horizontal asymptotes, logarithmic functions have vertical asymptotes, and both can be analyzed through standard transformations (stretches, reflections, and shifts).