A reliable method
- Identify the starting amount and repeated factor.
- Use an exponent to count equal steps.
- For logarithms require a positive input and positive base other than 1.
- Convert between logarithmic and exponential statements to check answers.
Example 1
A population model is \(P=200*2**x\). Find P at x=3.
- The starting quantity is 200.
- Three doubling steps give factor 2 cubed, or 8.
- 200 times 8 is 1,600.
Answer: \(1600\)
Example 2
Find \(\log_2 32\).
- Ask which exponent on 2 gives 32.
- 2 times itself five times is 32.
- Therefore the logarithm is 5; check 2 to the power 5 = 32.
Answer: \(5\)
Common mistakes
- A 6 percent increase uses factor 1.06, not 0.06.
- Logarithms do not distribute over addition.
Practice without the answer first
- A value of 500 falls by 20% per step. Find it after two steps.
- Find \(\log_3(1/27)\).
- A value starts at 100 and grows 10% per step. Find it after three steps.
Answers and step-by-step explanations
Practice 1
- Each step keeps 1-0.20=0.80 of the previous value.
- Two steps multiply by 0.80 squared.
- 500 times 0.80 squared is 320, not 300.
Answer: \(320\)
Practice 2
- 27 is 3 cubed.
- Its reciprocal is 3 to the power -3.
- The requested exponent is -3.
Answer: \(-3\)
Practice 3
- Each step multiplies by 1.1.
- The total factor is 1.1 cubed.
- 100 times 1.1 cubed is 133.1.
Answer: \(1331/10\)
These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.
