Exponentials & logarithms
Growth by multiplying, and logs that answer “what power?”
Pen and paper is fine · no calculator needed why?
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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
- Write the number as a power of the same base.
- Set the two exponents equal.
- Solve that small equation for x.
worked example
Solve 2ˣ⁻¹ = 64.
- 2⁶ = 64, so x − 1 = 6.
- Add 1: x = 7.
Answer: 7
Combine the logs
- Adding logs multiplies the insides. Subtracting logs divides them.
- A number in front becomes a power: 2 log₃ 5 is log₃ 25.
- Once it is one log, ask what power of the base gives the inside.
worked example
What is log₂ 40 − log₂ 5?
- Subtracting logs divides the insides: 40 ÷ 5 = 8.
- 2³ = 8, so the answer is 3.
Answer: 3
Multiply by the growth factor
- Turn the rate into a growth factor: 10% becomes 1.1, and 5% becomes 1.05.
- Multiply by the factor once for each year.
- Keep every cent until the end.
worked example
$1,000 grows 20% per year, compounded yearly. What is it worth after 3 years?
- The growth factor is 1.2.
- Year 1: $1,000 × 1.2 = $1,200.
- Year 2: $1,200 × 1.2 = $1,440.
- Year 3: $1,440 × 1.2 = $1,728.
Answer: $1,728
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.
practice
Solve for the exponent
worked example
Solve 2⁴ˣ = 16.
Answer: 1
- 16 = 2⁴, so 4x = 4 and x = 1.
Evaluate a log
worked example
What is log₂ 4?
Answer: 2
- 2² = 4, so log₂ 4 = 2.
Compound growth
worked example
$9,800 grows 20% per year, compounded yearly. What is it worth after 3 years?
Answer: $16,934.40
- Multiply by 1.2 three times: $11,760, then $14,112, then $16,934.40.
- Check: $9,800 × 1.2³ = $16,934.40.
Log rules
worked example
What is log₃ 45 − log₃ 5?
Answer: 2
- Subtracting logs divides the insides: log₃ (45 ÷ 5) = log₃ 9.
- 3² = 9, so the answer is 2.