Exponentials and Logarithms

Learn exponential growth, decay, logarithms, equations, and models.

Learning objectives

After this lesson you should be able to:

  • evaluate positive, zero, and negative exponents;
  • distinguish exponential growth from exponential decay;
  • interpret parameters in a*b^x and A(1 + r)^t;
  • explain logarithms as inverse operations;
  • convert between logarithmic and exponential form;
  • solve basic exponential and logarithmic equations;
  • check logarithm domain restrictions;
  • use SymPy and browser symbolic tools to verify your work.

Motivation: repeated multiplication

Linear growth adds the same amount each step. Exponential growth multiplies by the same factor each step. If $100 grows by 5% per year, the values are:

100, 100(1.05), 100(1.05)^2, 100(1.05)^3, ...

Exponential functions appear in compound interest, population growth, medicine decay, half-life, and many risk models.

Exponent rules with meaning

Rule Example Meaning
a^m * a^n = a^(m+n) 2^3 * 2^4 = 2^7 combine repeated factors
(a^m)^n = a^(mn) (2^3)^4 = 2^12 repeat a repeated product
a^0 = 1 5^0 = 1 keeps exponent patterns consistent
a^-n = 1/a^n 2^-3 = 1/8 negative exponent means reciprocal

Common mistake

2^3 + 2^4 is not 2^7. Exponent rules apply to multiplication with the same base, not addition.

Exponential functions

An exponential function has the variable in the exponent:

f(x) = a*b^x
  • a is the starting value when x = 0;
  • b is the growth or decay factor;
  • b > 1 gives growth;
  • 0 < b < 1 gives decay.

Example: P(t) = 200(1.04)^t starts at 200 and grows by 4% per time step. Example: A(t) = 80(0.75)^t keeps 75% and loses 25% per time step.

Percent growth and decay

The model:

A(t) = A_0(1 + r)^t

uses a rate r written as a decimal. A 6% growth rate gives factor 1.06. A 20% decay rate gives factor 0.80.

Key idea

A growth rate is not the same as a growth factor. Add the rate to 1 for growth and subtract it from 1 for decay.

Logarithms undo exponentials

A logarithm answers: what exponent produced this number?

log_b(y) = x means b^x = y
Exponential form Logarithmic form
2^3 = 8 log_2(8) = 3
10^2 = 100 log_10(100) = 2
e^0 = 1 ln(1) = 0

The natural logarithm ln(x) means log_e(x).

Domain and range

For b^x with b > 0 and b != 1, the domain is all real numbers and the range is positive real numbers. For log_b(x), the domain is positive real numbers and the range is all real numbers.

Thus log(x - 4) requires x - 4 > 0, so x > 4.

Log domain warning

In real-valued algebra, log(0) and log(-3) are not defined. Always check log inputs before accepting a solution.

Logarithm laws

For M > 0, N > 0, b > 0, and b != 1:

log_b(MN) = log_b(M) + log_b(N)
log_b(M/N) = log_b(M) - log_b(N)
log_b(M^p) = p log_b(M)

Logarithms do not distribute over addition: log(x + y) is not log(x) + log(y).

Solving exponential equations

Common-base example:

2^(x + 1) = 16 = 2^4
x + 1 = 4
x = 3

Taking logarithms:

3^x = 20
log(3^x) = log(20)
x log(3) = log(20)
x = log(20)/log(3)

Growth model:

100(1.05)^t = 150
(1.05)^t = 1.5
t = log(1.5)/log(1.05)

Solving logarithmic equations

Direct rewrite:

log_2(x - 1) = 3
2^3 = x - 1
x = 9

Check the domain: x > 1, so 9 is valid.

Combining logs:

log(x) + log(x - 3) = log(10)

Domain: x > 3. Combine:

log(x(x - 3)) = log(10)
x(x - 3) = 10
x^2 - 3x - 10 = 0
(x - 5)(x + 2) = 0

Only x = 5 satisfies the domain.

Applications

  • Compound interest: A(t) = 500(1.06)^t.
  • Half-life: A(t) = A_0(1/2)^(t/h) where h is the half-life.
  • Doubling time: P(t) = P_0*2^(t/d) where d is the doubling time.

Browser symbolic practice

Useful browser tools include math.js, Nerdamer, Algebrite, and PyScript. Try evaluating an exponential model, comparing 2^x and x^2, solving 3^x = 20, and checking domains such as log(2x - 5).

Practice exercises

  1. Evaluate 2^5 and 2^-3.
  2. Does 300(1.07)^t represent growth or decay? What is the percent rate?
  3. Does 90(0.82)^t represent growth or decay? What is the percent rate?
  4. Rewrite log_5(125) = 3 in exponential form.
  5. Rewrite 4^3 = 64 in logarithmic form.
  6. Solve 2^(x - 1) = 32.
  7. Solve 5^x = 18 exactly.
  8. Find the domain of log(x + 6).
  9. Solve log_3(x + 2) = 4.
  10. Write a model for a value starting at 1000 and decaying by 12% per year.
  1. 32 and 1/8.
  2. Growth by 7%.
  3. Decay by 18%.
  4. 5^3 = 125.
  5. log_4(64) = 3.
  6. x = 6.
  7. x = log(18)/log(5).
  8. x > -6.
  9. x = 79.
  10. A(t) = 1000(0.88)^t.

Subtopic guided practice and checkpoints

Exponentials build repeated multiplication. Logarithms answer exponent questions. Guess which inverse operation you need before solving.

Lab 1: exponent rules with meaning

Guess first. Is a^3 * a^4 closer to a^7 or a^12?

Guided exercise.

a^3 * a^4 = (a*a*a)(a*a*a*a) = a^7

Add exponents only when the bases match and you are multiplying powers.

Checkpoint. Simplify x^5/x^2 and (2x^3)^2. State which rule you used.

Lab 2: growth and decay

Guess first. In P(t) = 500(1.08)^t, is the quantity growing or decaying?

Guided exercise.

The base is 1.08, which is larger than 1, so the model grows by 8% each time period. At t = 3:

P(3) = 500(1.08)^3
     = 629.856

Checkpoint. Interpret A(t) = 1200(0.85)^t. Is it growth or decay? What is the percent change each time period?

Lab 3: logarithms as inverse questions

Guess first. What power of 2 gives 32?

Guided exercise.

log_2(32) = ?
2^? = 32
2^5 = 32
log_2(32) = 5

Checkpoint. Rewrite log_3(81) = 4 in exponential form. Then evaluate log_5(1/25).

Lab 4: solve exponential and logarithmic equations

Guess first. To solve 3^(x - 1) = 27, should you use common bases or logs?

Guided exercise.

3^(x - 1) = 27
3^(x - 1) = 3^3
x - 1 = 3
x = 4

For a logarithmic equation, rewrite in exponential form:

log_2(x + 6) = 5
x + 6 = 2^5
x + 6 = 32
x = 26

Check the log domain: x + 6 > 0, and 26 works.

Checkpoint. Solve 5^(2x) = 125 and log_3(x - 1) = 2. Include a domain check for the logarithmic equation.

Guessing game checkpoint

Using this lesson with edumath and SymPy

Use parsed equations with the solver for exact symbolic answers, then interpret them using exponent and logarithm rules.

from edumath.core import parse_equation
from edumath.solvers import solve_equation_steps

exponential_solution = solve_equation_steps(parse_equation("3^(x - 1) = 27"))
print(exponential_solution.render_text())
Answer: x = 4

Method: SymPy solveset

Steps:

1. Original equation:
Eq(3**(x - 1*1), 27)

2. Solve symbolically:
x ∈ {4}
SymPy found the solution set directly.

Check:

x = 4: valid
import sympy as sp

x = sp.Symbol("x", positive=True)
sp.solve(sp.Eq(3**x, 20), x)
[log(20)/log(3)]
sp.solve(sp.Eq(sp.log(x, 10), 3), x)
[1000]
equation = sp.Eq(sp.log(x) + sp.log(x - 3), sp.log(10))
candidates = sp.solve(equation, x)
[(candidate, sp.checksol(equation, x, candidate)) for candidate in candidates]
[(5, True)]
t = sp.symbols("t")
solution = sp.solve(sp.Eq(100 * 1.05**t, 150), t)[0]
float(solution)
8.310386222520568

Further reading