doing math

courses › Algebra

Exponentials & logarithms

level 30 course

Growth by multiplying, and logs that answer “what power?”

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 30.

Sign in to start

Builds on: Functions (not open yet) · Exponents & roots (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

An exponent counts repeated multiplying: 3⁴ = 3 × 3 × 3 × 3 = 81. A logarithm asks the reverse question, what power? log₃ 81 asks what power of 3 gives 81, and the answer is 4. A fraction such as 1/9 is a negative power of 3, so log₃ (1/9) is −2.

Growth works by multiplying too. Growing 10% a year means multiplying by 1.1 each year, not adding the same amount, so the growth itself grows. That is compounding.

Techniques

Match the powers

Solving an equation like 2ˣ⁻¹ = 64 or 3²ˣ = 81.

  1. Write the number as a power of the same base.
  2. Set the two exponents equal.
  3. Solve that small equation for x.
worked example

Example: Solve 2ˣ⁻¹ = 64.

  1. 2⁶ = 64, so x − 1 = 6.
  2. Add 1: x = 7.

Answer: 7

Combine the logs

Logs with the same base that are added, subtracted, or have a number in front.

  1. Adding logs multiplies the insides. Subtracting logs divides them.
  2. A number in front becomes a power: 2 log₃ 5 is log₃ 25.
  3. Once it is one log, ask what power of the base gives the inside.
worked example

Example: What is log₂ 40 − log₂ 5?

  1. Subtracting logs divides the insides: 40 ÷ 5 = 8.
  2. 2³ = 8, so the answer is 3.

Answer: 3

Multiply by the growth factor

An amount grows by the same percent each year, compounded yearly.

  1. Turn the rate into a growth factor: 10% becomes 1.1, and 5% becomes 1.05.
  2. Multiply by the factor once for each year.
  3. Keep every cent until the end.
worked example

Example: $1,000 grows 20% per year, compounded yearly. What is it worth after 3 years?

  1. The growth factor is 1.2.
  2. Year 1: $1,000 × 1.2 = $1,200.
  3. Year 2: $1,200 × 1.2 = $1,440.
  4. Year 3: $1,440 × 1.2 = $1,728.

Answer: $1,728

Tips by skill

  • TipSolve for the exponent: Write the number as a power of the same base, set the exponents equal, and solve for x.
  • TipEvaluate a log: Ask what power of the base gives the number. A fraction like 1/25 needs a negative power, and 1 needs the power 0.
  • TipCompound growth: Multiply by 1 plus the rate as a decimal, once for each year. Never add a fixed amount each year.
  • TipLog rules: Combine into one log first: adding multiplies the insides, subtracting divides them, and a number in front becomes a power.

Watch out for

  • Finding the power but stopping there. In 2ˣ⁻¹ = 64 the power is 6, so x is 7, not 6.
  • Reading a log as division. log₃ 81 asks for a power of 3, not 81 ÷ 3.
  • Subtracting the insides of two logs instead of dividing them.
  • Adding the same amount each year. At 10% a year, $1,000 grows to $1,210 after two years, not $1,200.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

your rounds

No rounds yet.