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 the inverse
A logarithm undoes an exponential, so their graphs reflect across y = x.
Domain and range swap
An exponential is always positive, 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.
What they do not do
The log of a sum is not the sum of the logs. Only products and quotients split.
A logarithm asks what power
log_b y is the exponent giving y from base b. Reading every log statement back as an exponential statement is the most useful habit available in this chapter.
The laws mirror the exponent laws
Products become sums, quotients become differences, and a power comes down as a coefficient. Each is an exponent law seen through the inverse, so none is a separate fact.
The power rule is the operative one
log(aⁿ) = n log a brings an exponent down to the front, which is exactly what allows equations with the unknown in the exponent to be solved. It is why logarithms were invented.
What the laws do not say
log(a + b) is not log a + log b. The laws convert products to sums, never sums to anything. Inventing a rule for sums is the standard and complete error in the topic.
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 Combine Logs
Ines simplifies log 4 + log 25.
- Step 1
The product law says a sum of logs is the log of the product.
Step 4: Your Turn
Practice makes it stick.
The Definition
Problem 1 of 2
What is log base 2 of 8?
The Product
Problem 2 of 2
log 4 + log 25 equals log of what number?
Apply the Laws
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 12 − log 3 equals log of what number?
5 of 8
log(x⁵) 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 64?
Step 5: Quick Check
Show what you know.
Question 1 of 2
What is log base 4 of 16?
Question 2 of 2
Is the log of a sum the sum of the logs?
What You Learned
- A logarithm answers what exponent produces a value.
- It is the inverse of an exponential, so the domain is positive numbers only.
- The laws are the exponent rules read backwards.