9.5 Properties of Logarithms - Algebra 2 - Free Printable

Let’s solve this step by step.

We are given:

6.2(log₄ 6 − log₄ 3) + 0.5 log₄ 25

Wait — looking at the handwritten work, it seems like the “6.2” might actually be a typo or miswriting of “2”, because in the student’s work they wrote:

> 2 log₄ 6 − 2 log₄ 3 + log₄ 25^(1/2)

And then simplified to:

> log₄ 36 − log₄ 9 + log₄ 5

Which suggests that the original problem was likely:

2(log₄ 6 − log₄ 3) + 0.5 log₄ 25

Because:
- 2 × log₄ 6 = log₄(6²) = log₄ 36
- 2 × log₄ 3 = log₄(3²) = log₄ 9
- 0.5 × log₄ 25 = log₄(25^0.5) = log₄ 5

So we’ll proceed with 2(log₄ 6 − log₄ 3) + 0.5 log₄ 25, since that matches the student’s steps and makes sense mathematically.

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Step 1: Use logarithm rules

Recall these key log rules:

1. a·log_b(x) = log_b(x^a) → power rule
2. log_b(x) − log_b(y) = log_b(x/y) → quotient rule
3. log_b(x) + log_b(y) = log_b(x·y) → product rule

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Step 2: Apply the power rule to each term

First part:
2(log₄ 6 − log₄ 3)
= 2·log₄ 6 − 2·log₄ 3
= log₄(6²) − log₄(3²)
= log₄ 36 − log₄ 9

Second part:
0.5 log₄ 25
= log₄(25^0.5)
= log₄ √25
= log₄ 5 (since √25 = 5)

Now our expression is:

log₄ 36 − log₄ 9 + log₄ 5

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Step 3: Combine using quotient and product rules

Start with:
log₄ 36 − log₄ 9 = log₄(36/9) = log₄ 4

Then add log₄ 5:
log₄ 4 + log₄ 5 = log₄(4 × 5) = log₄ 20

Wait — but in the student’s work, they got:

> log₄ 4 + log₄ 5 = 1 + log₄ 5

That’s also correct! Because:

log₄ 4 = 1 (since 4¹ = 4)

So both forms are valid:

→ log₄ 20
OR
→ 1 + log₄ 5

But which one is “condensed”? Usually, condensed means written as a single logarithm if possible.

log₄ 20 is a single log — so that’s more condensed.

But let’s check: Is log₄ 20 equal to 1 + log₄ 5?

Yes:

1 + log₄ 5 = log₄ 4 + log₄ 5 = log₄(4×5) = log₄ 20 ✔

So both are correct, but log₄ 20 is fully condensed into one log term.

However, looking back at the student’s final box: they wrote 1 + log₄ 5, which is also acceptable — sometimes teachers accept either form.

But since the instruction says “Condense”, and condensing usually means combining into a single logarithmic expression, log₄ 20 is better.

Wait — let me double-check the arithmetic.

Original after applying powers:

log₄ 36 − log₄ 9 + log₄ 5

36 ÷ 9 = 4 → log₄ 4 + log₄ 5 = log₄(4×5) = log₄ 20 ✔️

Alternatively, if you leave it as 1 + log₄ 5, that’s not a single log — it’s a sum.

So for full condensation, log₄ 20 is the best answer.

But let’s see what the student did — they stopped at 1 + log₄ 5, which is fine, but maybe not fully condensed.

Actually, let’s re-read the problem: It says “Condense”. In many textbooks, “condense” means write as a single logarithm.

Example: Condense log A + log B → log(AB)

So here, we should go all the way to log₄ 20

But wait — is there any simplification beyond that? 20 doesn’t have a nice power of 4, so log₄ 20 is simplest single-log form.

Alternatively, if the problem allows mixed forms, 1 + log₄ 5 is also correct and perhaps simpler numerically.

But again — “condense” typically means combine into one log.

Let me verify with numbers to make sure.

Compute numerical value:

log₄ 20 = ln20 / ln4 ≈ 2.9957 / 1.3863 ≈ 2.161

1 + log₄ 5 = 1 + (ln5 / ln4) ≈ 1 + (1.6094 / 1.3863) ≈ 1 + 1.161 ≈ 2.161 → same ✔️

So both are equivalent.

But since the instruction is “Condense”, and the student’s work ended with 1 + log₄ 5, which is partially condensed, I think the expected fully condensed answer is log₄ 20

However, looking at the yellow highlight on “5” and the box around “1 + log₄ 5”, it seems the teacher or system accepted that form.

To be safe, let’s present both and choose the most appropriate.

In most algebra classes, when asked to “condense”, they want a single logarithm.

For example, from Khan Academy or standard texts:

“Condense the expression into a single logarithm.”

So I will go with log₄ 20

But let’s check once more:

Original: 2(log₄ 6 − log₄ 3) + 0.5 log₄ 25

= 2 log₄(6/3) + log₄(25^{0.5}) [using quotient inside first]

= 2 log₄ 2 + log₄ 5

= log₄(2²) + log₄ 5

= log₄ 4 + log₄ 5

= log₄(4×5) = log₄ 20

Same result.

If we do it this way, even faster:

Inside the parentheses: log₄ 6 − log₄ 3 = log₄(6/3) = log₄ 2

Then multiply by 2: 2 log₄ 2 = log₄ 4

Then 0.5 log₄ 25 = log₄ 5

Then log₄ 4 + log₄ 5 = log₄ 20

Perfect.

So final condensed form is log₄ 20

But why did the student write 1 + log₄ 5? Probably because they recognized log₄ 4 = 1, and left it like that. It’s not wrong, but not fully condensed.

Since the problem says “Condense”, and doesn’t specify “as much as possible” or “into a single log”, but in context, single log is standard.

I think the safest and most correct answer for “condense” is log₄ 20

Final Answer:
log₄ 20