Exponential & logarithmic functions

Exponential and logarithmic functions are inverse ideas. They model fast growth, rapid decay, and scale changes that linear models cannot describe well.

Exponential basics

An exponential function has variable in the exponent: \[ y=a^x,\quad a>0,\ a\ne1. \]

If \(a>1\), the function grows as \(x\) increases. If \(0<a<1\), it decays.

What is a logarithm?

A logarithm asks for the exponent: \[ \log_a b = c \iff a^c=b. \]

So \(\log_2 8=3\) because \(2^3=8\). Also, \(\log_{10}100=2\).

Domain and range

• \(y=a^x\): domain all real \(x\), range \(y>0\), horizontal asymptote \(y=0\).
• \(y=\log_a x\): domain \(x>0\), range all real numbers, vertical asymptote \(x=0\).

These two graphs are reflections of each other in the line \(y=x\).

Index laws review

Exponential solving often uses:

\[ a^m a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn}. \]

Logarithm laws

For \(M,N>0\):

\[ \log_a(MN)=\log_a M+\log_a N, \] \[ \log_a\!\left(\frac{M}{N}\right)=\log_a M-\log_a N, \] \[ \log_a(M^k)=k\log_a M. \]

Common mistakes: \(\log(M+N)\ne \log M+\log N\).

Change of base

To evaluate logs with calculator: \[ \log_a b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}. \]

Here \(\log\) usually means base 10 and \(\ln\) means base \(e\).

Solving exponential equations

Case 1: same base.

\[ 3^{2x-1}=3^5 \Rightarrow 2x-1=5 \Rightarrow x=3. \]

Case 2: take logs.

\[ 5^x=17 \Rightarrow x=\log_5 17=\frac{\log 17}{\log 5}\approx 1.760. \]

Solving logarithmic equations

Example 1. \(\log_2(x-1)=4\)

Convert to exponential: \(x-1=2^4=16\), so \(x=17\). Domain check: \(x>1\), valid.

Example 2. \(\log_{10}(x+3)+\log_{10}(x-1)=1\)

\[ \log_{10}\big((x+3)(x-1)\big)=1 \Rightarrow (x+3)(x-1)=10. \] \[ x^2+2x-3=10 \Rightarrow x^2+2x-13=0. \] \[ x=-1\pm\sqrt{14}. \] Domain requires \(x>1\), so \(x=-1+\sqrt{14}\) only.

Growth and decay models

A common model: \[ A(t)=A_0(1+r)^t \] for growth (\(r>0\)) or decay (\(r<0\), with \(1+r>0\)).

Continuous model: \[ A(t)=A_0e^{kt}, \] where \(k>0\) gives growth and \(k<0\) gives decay.

Exam strategy

• State domain restrictions first in log equations.
• When combining logs, check arguments stay positive.
• Use exact forms before calculator rounding.
• For modeling, define variables and units clearly.

History

Logarithms were developed in the early 17th century (notably by John Napier) to simplify long multiplication and division into addition and subtraction. Before electronic calculators, this was a major speed boost for astronomy, navigation, and engineering.

Exponential functions became central in science through models of population growth, radioactive decay, and continuously changing quantities.

Derivation and reasoning

Why log product law works

Let \(\log_a M=p\) and \(\log_a N=q\). Then \(M=a^p\), \(N=a^q\), so \[ MN=a^{p+q}. \] Taking \(\log_a\) gives: \[ \log_a(MN)=p+q=\log_a M+\log_a N. \]

Inverse relationship view

If \(y=a^x\), swapping \(x,y\) gives \(x=a^y\), so \(y=\log_a x\). That is why exponential and logarithmic graphs are mirror images across \(y=x\).

Checkpoints

Evaluate \(\log_3 81\) and \(\log_{10}0.01\).

Answer: \(\log_3 81=4\), \(\log_{10}0.01=-2\).

Expand: \(\log_2(8x^3)\).

Answer: \(\log_2 8 + \log_2 x^3 = 3 + 3\log_2 x\).

Solve \(2^{x+1}=7\).

Answer: \(x+1=\log_2 7\), so \(x=\log_2 7-1 \approx 1.807\).

Solve \(\log_5(x-4)=2\) and state domain condition.

Domain: \(x-4>0\Rightarrow x>4\).

Answer: \(x-4=25\Rightarrow x=29\), valid.

A population starts at 500 and grows 6% per year. Write the model and find \(A(4)\).

Model: \(A(t)=500(1.06)^t\).

Value: \(A(4)=500(1.06)^4\approx 631.24\), about 631.

A substance follows \(A(t)=120e^{-0.18t}\). Is this growth or decay? Why?

Answer: Decay, because exponent coefficient is negative (\(k=-0.18\)).

Applications

Compound interest

Savings and loans often follow exponential rules. Logarithms are used to solve for unknown time or interest rate.

Half-life and decay

Radioactive substances and medicine concentration in the body are modeled with exponential decay.

Logarithmic scales

pH, decibels, and Richter magnitude use logarithmic scales to represent very large ratio ranges in manageable numbers.

References

• Standard high-school curricula on exponential and logarithmic functions (IGCSE, IB, A-Level, AP).
• Precalculus and algebra references on log laws, inverse functions, and modeling.