Packets  /  Exponential and Logarithmic Functions
Grades 9-12Algebra

Exponential and Logarithmic Functions

Exponential and logarithmic functions from the ground up: exponent rules, graphs, solving exponential and logarithmic equations, log properties, the natural log, and real-world growth and decay models. Learn each concept, practice with instant feedback, and beat the 60-second challenge.

1Exponent Rules

Key idea: Five rules handle every simplification: multiply powers add exponents, divide powers subtract exponents, power of a power multiplies exponents, negative exponents flip into fractions, and any nonzero base to the zero power is 1.
  1. Product: xa · xb = xa+b.
  2. Quotient: xa / xb = xa-b.
  3. Power of a power: (xa)b = xa·b.
  4. Negative: x-n = 1 / xn.
  5. Zero: x0 = 1 (as long as x is not 0).
Worked example
x3 · x4
Multiplying with the same base: add the exponents. 3 + 4 = 7, so x7.
Worked example
(x2)3
Power of a power: multiply the exponents. 2 · 3 = 6, so x6.
Watch out: The #1 mistake: x^{3} · x^{4} = x^{12} is wrong — you ADD exponents when multiplying! You only multiply exponents for a power of a power.

2Exponential Functions and Graphs

Key idea: f(x) = b^{x} with b > 0 and b not 1. If b > 1 the graph grows; if 0 < b < 1 it decays. Every exponential graph passes through (0, 1), and the x-axis is a horizontal asymptote the graph never touches.
  1. Identify b, the base.
  2. b > 1 means exponential growth; 0 < b < 1 means exponential decay.
  3. The y-intercept is always (0, 1), because b0 = 1.
  4. To evaluate, substitute the x-value and compute.
  5. To sketch, plot (0, 1) and a couple more points, then draw the smooth curve approaching the x-axis on one side.
Worked example
f(x) = 2x, find f(3)
Growth (base 2 > 1). f(3) = 23 = 8.
Worked example
f(x) = 12x, find f(-1)
Decay (base 1/2 < 1). f(-1) = 12-1 = 2.
Watch out: A negative exponent does not make the answer negative: 2^{-3} = 1/8, still positive! And watch out: -2^{x} is not the same as (-2)^{x}.

3Solving Exponential Equations

Key idea: If the bases match, the exponents must match: b^{M} = b^{N} means M = N. Rewrite both sides as powers of the same base, then solve the exponent equation.
  1. Isolate the exponential part — divide away any coefficient first.
  2. Rewrite both sides with the same base.
  3. Set the exponents equal and solve for x.
  4. Check by substituting back into the original equation.
Worked example
2x+1 = 25
Same base 2, so x + 1 = 5. Therefore x = 4.
Worked example
3 · 2x = 48
Divide by 3 first: 2x = 16 = 24, so x = 4.
Watch out: You can only equate exponents when the bases are identical! And never combine a coefficient into the base: in 3 · 2^{x} = 48, writing 6^{x} = 48 is wrong.

4What Logarithms Mean

Key idea: log_b(a) = c means b^{c} = a. A logarithm asks: to what power must I raise the base to get the argument?
  1. Name the base b and the argument a.
  2. Ask: b to what power equals a?
  3. Convert freely: log_b(a) = c means bc = a.
Worked example
log_2(8)
Ask: 2 to what power is 8? 23 = 8, so the answer is 3.
Worked example
Write 102 = 100 in logarithmic form
log_10(100) = 2.
Watch out: log_b(1) = 0 always, because b^{0} = 1. But log of 0 or a negative number does not exist!

5Properties of Logarithms

Key idea: Logs turn multiplication into addition, division into subtraction, and powers into multiplication — but every log must share the same base.
  1. Product: log_b(x) + log_b(y) = log_b(xy).
  2. Quotient: log_b(x) – log_b(y) = log_b(x/y).
  3. Power: n · log_b(x) = log_b(xn).
Worked example
log_2(3) + log_2(5)
Product rule: log_2(3 · 5) = log_2(15).
Worked example
4 · log_3(2)
Power rule: log_3(24) = log_3(16).
Watch out: log(x + y) is NOT log x + log y! The product rule only splits multiplication inside the log — never addition.

6Expanding and Condensing Logs

Key idea: Expanding runs the log rules forward to break one log apart; condensing runs them backward to fuse logs into a single log. Exponents drop down as coefficients; multiplication becomes +, division becomes -.
  1. Spot the structure: power? product? quotient?
  2. To expand: bring exponents down front, then split products (× → +) and quotients (÷ → -).
  3. To condense: push coefficients back up as exponents, then fuse into one log.
Worked example
Expand: log_3(x4)
Bring the exponent down front: 4 · log_3(x).
Worked example
Condense: 2 · ln(x) + 3 · ln(y)
Push coefficients up: ln(x2) + ln(y3), then fuse: ln(x2 y3).
Watch out: A minus sign distributes: log_b(x) – 2 · log_b(y) = log_b(x / y^{2}). And you can never distribute a log over + or -!

7Solving Logarithmic Equations

Key idea: Convert to exponential form and solve — then CHECK! Any solution that makes a logarithm's argument zero or negative is extraneous: throw it out.
  1. Isolate the logarithm (combine multiple logs into one first).
  2. Rewrite exponentially: log_b(x) = c means x = bc.
  3. Solve for x.
  4. CHECK every answer in the original equation.
Worked example
log_3(x) = 4
Rewrite exponentially: x = 34 = 81. Check: log_3(81) = 4.
Worked example
log_2(x) – log_2(5) = 3
Combine: log_2(x/5) = 3, so x/5 = 8 and x = 40. Check: log_2(40) – log_2(5) = log_2(8) = 3.
Watch out: Skipping the check is the #1 error! If solving gives x = -3 in log_2(x) = 5, reject it — log_2(-3) does not exist.

8The Natural Log and e

Key idea: e is about 2.718, nature's favorite base, and ln(x) means log_e(x). Two shortcuts do most of the work: ln(e^{k}) = k and e^{ln(x)} = x.
  1. See ln(e…)? ln and e cancel: ln(ek) = k.
  2. See eln(…)? They cancel the other way: eln(x) = x.
  3. Memorize: ln(1) = 0 and ln(e) = 1.
Worked example
ln(e7)
ln and e undo each other, so the answer is 7.
Worked example
Solve: ex = e-2
Same base e, so x = -2.
Watch out: ln(0) is undefined, and e^{ln(x)} = x needs x > 0. Also ln(e + x) does not simplify — no shortcut for sums!

9Applications: Growth, Decay and Interest

Key idea: Exponentials model real growth: compound interest A = P(1 + r)^{t}, population growth y = a · b^{t}, half-life decay y = a(1/2)^{t/h}. Turn percents into decimals, match time units, round money to cents.
  1. Name P (starting amount), r (rate as a decimal), t (time).
  2. Pick the model: interest P(1+r)t, growth a·bt, half-life a12t/h.
  3. Substitute and compute step by step.
Worked example
$1,000 at 5% compounded yearly for 2 years
A = 1000(1.05)2 = 1000(1.1025) = $1,102.50.
Worked example
240 g sample halves every 3 hours — after 9 hours?
9 ÷ 3 = 3 halvings: 240 → 120 → 60 → 30 g.
Watch out: 5% means r = 0.05, not 5! And compounded yearly counts t in years — always match time units to the rate.
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60-Second Challenge

How many exponential and logarithmic functions problems can you solve in 60 seconds?