Question

Use properties of logs to simplify the expression

Answer 1

\[\log_{9}(x-\sqrt{x ^{2}-45})+ \log_{9}(x+\sqrt{x ^{2}-45})\]

Answer 2

so what I got was \[\log_{9}45 \] but that's not one of my answer choices. what am I doing wrong?

Answer 3

these are the answer choices:
1. 1 + log9 5
2. log5 9
3. 1 + log5 9
4. log9 5
5. 9 + log9 5

Answer 4

The sum of logs is the log of the product.

\( \log_x a + \log_b y = \log_b xy \)

Answer 5

\(\log_{9}(x-\sqrt{x ^{2}-45})+ \log_{9}(x+\sqrt{x ^{2}-45}) \)

\( = \log_{9}[(x-\sqrt{x ^{2}-45})(x+\sqrt{x ^{2}-45})] \)

Now use "the product of a sum and a difference equals the difference of two squares" pattern to simplify.

Answer 6

45?

Answer 7

\( = \log_9 45\)

You are correct, but you can go furhter because 45 = 9 * 5.

Answer 8

oh ok. I just need to know how to do the next step.

Answer 9

\( = \log_9 (9 \times 5) \)

Use the rule: log of product = sum of logs.

Answer 10

\( = \log_9 9 + \log_9 5 \)

\(\log_9 9\) can be simplified since \( \log_b b = 1\)

Answer 11

so \[1+\log_{9}5 \]

Answer 12

Exactly. Good job!