A logarithm answers "what exponent gives this?". log₂8 = 3 because 2³ = 8.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
It is an inverse
The logarithm undoes the exponential, so their graphs are reflections across y = x.
Domain and range swap
An exponential is positive everywhere, so a logarithm accepts only positive inputs. Its range is every real number.
The three laws
The log of a product is a sum, of a quotient is a difference, and of a power brings the exponent out front.
Why they hold
They are the exponent rules in reverse. Multiplying powers adds exponents, so the log of a product adds logs.
Solving for an exponent
When the unknown sits in the exponent, take logs of both sides. The power law then brings it down to where algebra reaches it.
The question an exponential cannot answer
Given bˣ = y, solving for x requires the logarithm: x = log_b y. A logarithm is the exponent, and reading every log statement back as an exponential one is the most useful habit in the topic.
Inverse functions
The log and the exponential with the same base undo each other, so their graphs are reflections in y = x. Domain and range swap: the exponential's range of positives becomes the log's domain.
The laws mirror the exponent laws
log(ab) = log a + log b, log(a/b) = log a − log b, and log(aⁿ) = n log a. Each is an exponent law seen through the inverse, and the third is what lets you solve for an unknown exponent.
The domain is positive only
No power of a positive base gives zero or a negative, so logs of those are undefined. Solutions to log equations must be checked against this, and some legitimately have to be discarded.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.69)
Step 3: Watch an Example
One step at a time.
Watch Ines Solve for an Exponent
Ines solves 3ˣ = 20.
- Step 1
The unknown is in the exponent, so ordinary algebra cannot reach it.
Step 4: Your Turn
Practice makes it stick.
The Definition
Problem 1 of 2
What is log base 2 of 8?
The Product Law
Problem 2 of 2
log 4 + log 25 equals log of what number?
Work With Logs
1 of 8
log base 3 of 9?
2 of 8
log base 5 of 125?
3 of 8
log base 10 of 1?
4 of 8
log 8 minus log 2 equals log of what number?
5 of 8
log(x³) equals k times log x. What is k?
6 of 8
Can a logarithm take a negative input? 1 yes, 0 no.
7 of 8
Match each log law with the exponent rule behind it.
Tap a card on the left to start.
8 of 8
log base 2 of 32?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is log base 4 of 64?
Question 2 of 2
Why can a logarithm not take a negative input?
What You Learned
- 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.