Cogito
AP Precalculus · Chapter 5 · Lesson 2
Logarithms as Inverses
The question an exponential cannot answer.
12 problems · about 22 minutes · AP Precalculus 2.7, 2.8, 2.10
What this lesson teaches
The student defines logarithms as inverses of exponentials and applies the logarithm laws.
- A logarithm answers what exponent produces a given value.
- It is the inverse of an exponential, so domain and range swap.
- The log laws are the exponent rules read backwards.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is log base 4 of 64?
Answer 3
Why 3.
Why can a logarithm not take a negative input?
Answer An exponential output is always positive, so nothing maps back from a negative.
Why No exponent produces a negative result.
log base 3 of 9?
Answer 2
Why 3² = 9.
log base 5 of 125?
Answer 3
Why 5³ = 125.
log base 10 of 1?
Answer 0
Why Anything to the zero.
Build It Up
The same ideas with more to keep track of.
3 problemslog 8 minus log 2 equals log of what number?
Answer 4
Why 8 ÷ 2.
log(x³) equals k times log x. What is k?
Answer 3
Why The power law.
Can a logarithm take a negative input? 1 yes, 0 no.
Answer 0
Why An exponential is never negative.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each log law with the exponent rule behind it.
Answer Log of a product is a sum → Multiplying powers adds exponents; Log of a quotient is a difference → Dividing powers subtracts exponents; Log of a power brings the exponent out → A power of a power multiplies exponents
Why Each law mirrors one exponent rule.
log base 2 of 32?
Answer 5
Why 2⁵ = 32.
The Definition: What is log base 2 of 8?
Answer 3
Why 3.
The Product Law: log 4 + log 25 equals log of what number?
Answer 100
Why 100.