Logarithms and exponential equations
Treat a logarithm as an exponent and use it to recover an input in an exponential model.
What you will tackle
- Interpret logarithmic notation and inverse exponential relationships
- Use domains, transformations and logarithm rules
- Solve exponential equations and interpret growth-model inputs
A logarithm names the required exponent
log_b(a) = c means b^c = a, with b > 0, b ≠ 1 and a > 0 for real logarithms. Therefore log₂8 = 3 and log₁₀0.01 = −2. The graph of log_b(x) for b > 1 has domain x > 0, passes through (1,0) and increases; x = 0 is a vertical asymptote.
Use rules on products, not arbitrary sums
For positive inputs, log_b(MN) = log_b M + log_b N and log_b(M^k) = k log_b M. There is no corresponding rule turning log(M + N) into log M + log N. In A·b^t = C, t = log(C/A)/log b; a contextual number of complete periods may require rounding after solving.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Evaluate log₃81 and log₁₀0.001. Solve 5·2^t = 160, and state the real domain of log₂(x − 3).
Show a starting hint
Rewrite a logarithm as an exponential statement.
Compare with the explained answer
- 3^4 = 81 and 10^(−3) = 0.001.
- 2^t = 32 = 2^5, so t = 5.
- x − 3 must be positive, so x > 3.
Answer / response criteria: 4; −3; t = 5; x > 3.
Independent practice
Independent 1
Solve 3^x = 20 exactly using logarithms. Simplify log₂(8·4), and explain why log₂(8 + 4) is not 3 + 2.
Compare with the explained answer
- Taking logs gives x log 3 = log 20, so x = log 20/log 3.
- The product rule gives log₂32 = 5 = 3 + 2.
- For the sum, log₂12 is not log₂8 + log₂4; the product rule does not apply to addition.
Answer / response criteria: x = log 20/log 3; product logarithm 5; no sum rule applies.
Application practice
Application 1
An original population model is P(t) = 500(1.1)^t. Find the exact time to reach 1000, and the first whole year when the model reaches at least 1000. State a limitation.
Compare with the explained answer
- Divide by 500: 1.1^t = 2, giving t = log 2/log 1.1 ≈ 7.27.
- The first complete year meeting the threshold is year 8.
- The conclusion assumes constant 10% growth and model validity over that period.
Answer / response criteria: t = log 2/log 1.1 ≈ 7.27; year 8 for a whole-year check.
What can you explain now?
An original population model is P(t) = 500(1.1)^t. Find the exact time to reach 1000, and the first whole year when the model reaches at least 1000. State a limitation.
Compare with the explanation
t = log 2/log 1.1 ≈ 7.27; year 8 for a whole-year check.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide