Logarithm Proofs Practice- Precalculus Exercises
What You're Actually Learning When You Do Logarithm Proofs
Let's be clear: logarithm proofs aren't busy work. They're training your brain to see how mathematical expressions relate to each other. When you prove that log(ab) = log(a) + log(b), you're not just memorizing a formula—you're understanding why the relationship exists.
Most students rush through these problems to get to the next section. That's a mistake. The people who actually understand precalculus are the ones who can derive relationships, not just apply them.
The Core Logarithm Properties You Need to Prove
Every logarithm proof problem boils down to one or more of these fundamental relationships:
- Product Rule: logb(xy) = logb(x) + logb(y)
- Quotient Rule: logb(x/y) = logb(x) − logb(y)
- Power Rule: logb(xn) = n · logb(x)
- Change of Base: loga(x) = logb(x) / logb(a)
- Reciprocal Rule: logb(a) = 1 / loga(b)
Your job is to demonstrate that these statements are always true using the definition of logarithms and basic algebra.
How to Approach Logarithm Proof Problems
Here's the actual process that works:
Step 1: Write Down What You Know
If you're proving logb(xy) = logb(x) + logb(y), start by defining x and y in terms of the base b.
Let x = bm and y = bn
This is non-negotiable. You're translating the problem into exponential form, which gives you something concrete to work with.
Step 2: Apply the Definition
From x = bm, you get logb(x) = m
From y = bn, you get logb(y) = n
Step 3: Work the Left Side
xy = bm · bn = bm+n
Therefore, logb(xy) = m + n
Step 4: Substitute What You Found
m + n = logb(x) + logb(y)
Done. The proof is complete.
Worked Examples
Example 1: Prove the Quotient Rule
Statement: logb(x/y) = logb(x) − logb(y)
Proof:
Let x = bm and y = bn
Then logb(x) = m and logb(y) = n
x/y = bm / bn = bm−n
logb(x/y) = m − n = logb(x) − logb(y) ✓
Example 2: Prove the Power Rule
Statement: logb(xn) = n · logb(x)
Proof:
Let x = bm, so logb(x) = m
xn = (bm)n = bmn
logb(xn) = mn = n · m = n · logb(x) ✓
Example 3: Prove the Change of Base Formula
Statement: loga(x) = logb(x) / logb(a)
Proof:
Let loga(x) = y, so ay = x
Take logb of both sides:
logb(ay) = logb(x)
y · logb(a) = logb(x)
y = logb(x) / logb(a)
Substitute y back: loga(x) = logb(x) / logb(a) ✓
Practice Problems
Try these before checking the solutions. No peeking early.
Problem 1
Prove: logb(1) = 0
Hint: Remember that b0 = 1 for any base.
Problem 2
Prove: logb(b) = 1
Hint: What exponent turns b into b?
Problem 3
Prove: blogb(x) = x
Hint: This is the definition of logarithms in exponential form.
Problem 4
Prove: logb(a) = 1 / loga(b)
Hint: Start with logb(a) = y and solve for y.
Problem 5
Prove: logb(x) = logb(c) · logc(x)
Hint: Use the change of base formula twice.
Solutions
Solution 1
b0 = 1 for all b ≠ 0
Taking logb of both sides: logb(1) = 0 ✓
Solution 2
b1 = b
Taking logb of both sides: logb(b) = 1 ✓
Solution 3
Let logb(x) = y
By definition: by = x
Substitute y back: blogb(x) = x ✓
Solution 4
Let logb(a) = y
Then by = a
Taking loga of both sides: y · loga(b) = 1
y = 1 / loga(b)
Substitute y back: logb(a) = 1 / loga(b) ✓
Solution 5
Using change of base on logc(x):
logc(x) = logb(x) / logb(c)
Rearrange: logb(x) = logb(c) · logc(x) ✓
Common Mistakes That Cost You Points
These errors show up constantly in precalculus exams:
- Skipping the variable substitution. Students try to prove log(ab) = log(a) + log(b) by just writing both sides and claiming they're equal. That's not a proof—that's restating the problem.
- Assuming the conclusion. If your proof uses the thing you're trying to prove, you've accomplished nothing. Work from definitions and known facts only.
- Forgetting domain restrictions. Logarithms require positive arguments and bases ≠ 1. Your proofs should acknowledge this.
- Mixing up addition and multiplication. log(a + b) ≠ log(a) + log(b). This confusion costs people constantly.
- Not connecting the steps. Each line of your proof should flow logically from the previous one.
Quick Reference: Logarithm Properties
| Property | Formula | Key Strategy |
|---|---|---|
| Product Rule | logb(xy) = logb(x) + logb(y) | Convert to exponents, multiply, convert back |
| Quotient Rule | logb(x/y) = logb(x) − logb(y) | Convert to exponents, divide, convert back |
| Power Rule | logb(xn) = n · logb(x) | Convert to exponents, apply exponent to exponent |
| Change of Base | loga(x) = logb(x) / logb(a) | Take log of both sides of definition |
| Reciprocal | logb(a) = 1 / loga(b) | Solve the definition equation for the target |
How to Practice Effectively
Don't just read proofs—reproduce them from scratch. Here's what works:
- Cover the solution and write the proof yourself
- Check your work line by line
- Identify exactly where you got stuck
- Repeat until you can do it without help
One properly worked proof teaches you more than ten half-finished attempts. Quality over quantity here.
If you're still struggling, go back to the definition: logb(x) = y means by = x. That's the foundation everything else builds on. Master that relationship and every proof becomes manageable.