When the unknown sits in the exponent, take a logarithm of both sides. The power law then brings it down.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
When the bases match
If both sides can be written with the same base, equate the exponents directly. No logarithm is needed.
Logarithmic equations
Rewrite the equation in exponential form, or combine the logs first and then convert.
Combine before converting
Two separate logs cannot be converted directly. Use the laws to make a single logarithm first.
Check the domain
A logarithm needs a positive input, so any candidate making an argument zero or negative must be rejected.
Why the check matters
Combining logs can widen the domain, letting through a value the original equation never permitted.
Take logs to free the exponent
For 3ˣ = 20, take logarithms of both sides and use the power rule to bring x down. That single move converts an exponential equation into a linear one.
Exponentiate to free the argument
For log(x + 1) = 2, raise the base to both sides. The log and the exponential cancel, leaving an ordinary equation. The two techniques are mirror images.
Check against the log domain
Logarithms accept only positive arguments, so solutions making any argument zero or negative must be discarded. Exponentiating can produce such candidates, which makes the check mandatory.
Combine logs before solving
An equation with several log terms should be reduced to a single log using the laws before exponentiating. Exponentiating a sum of logs term by term is invalid and is the usual mistake.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 4)
Step 3: Watch an Example
One step at a time.
Watch Marcus Solve for an Exponent
Marcus 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 Same Base
Problem 1 of 2
2ˣ = 32. What is x?
The Log Equation
Problem 2 of 2
log base 2 of x = 4. What is x?
Solve Both Kinds
1 of 8
2ˣ = 64. What is x?
2 of 8
3ˣ = 81. What is x?
3 of 8
log base 3 of x = 2. What is x?
4 of 8
log base 5 of x = 3. What is x?
5 of 8
5ˣ = 1. What is x?
6 of 8
A candidate makes a log argument negative. Is it valid? 1 yes, 0 no.
7 of 8
Put the solving process for a log equation in order.
- 1Rewrite in exponential form.
- 2Solve the resulting equation.
- 3Reject any candidate that makes an argument non-positive.
- 4Use the laws to combine into a single logarithm.
8 of 8
10ˣ = 1000. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
4ˣ = 64. What is x?
Question 2 of 2
Why must log equation solutions be checked?
What You Learned
- Take logs when the unknown is in the exponent.
- Matching bases lets you equate exponents directly.
- Combine logs before converting, then check every argument is positive.