A)If logx=0.25, what is logx4? Rewrite using the power rule, then evaluate. (1 point)

Question

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Question 2
A)
Which of the following is equivalent to log9?

Option #1: log(3+3)

Option #2: 3log3

Option #3: 2log3

(1 point)
Option #$$
Question 3
A)If logh=35, what is logh3? Rewrite using the power rule, then evaluate. Write your answer as a fraction.(1 point)
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Question 4
A)Evaluate log100−−−√3.(1 point)
log100−−−√3=$$
Question 5
A)Find the value of log0.00001.(1 point)
log0.00001=$$

give me the exact answer in the simplest form

Answer 1

Let's solve each question one by one:

Question 1

If \( \log x = 0.25 \), we want to find \( \log x 4 \):

Using the properties of logarithms, we can use the power rule:

\[ \log x 4 = \log (4^{0.25}) = 0.25 \log 4 \]

Now we need to evaluate \( \log 4 \):

Since \( 4 = 2^2 \):

\[ \log 4 = \log (2^2) = 2 \log 2 \]

Plugging this back in, we find:

\[ \log x 4 = 0.25 \cdot 2 \log 2 = 0.5 \log 2 \]

Question 2

We want to determine which of the options is equivalent to \( \log 9 \):

1. \( \log(3 + 3) \) simplifies to \( \log(6) \), which is not equivalent to \( \log 9 \).
2. \( 3 \log 3 \) is equal to \( \log(3^3) = \log(27) \), which is not equivalent to \( \log 9 \).
3. \( 2 \log 3 \) is equal to \( \log(3^2) = \log(9) \), which is equivalent.

Therefore, the correct answer is Option #3: \( 2 \log 3 \).

Question 3

If \( \log h = 35 \), we want to find \( \log h 3 \).

Using the same method as before, we write:

\[ \log h 3 = \log(3^{35}) \]

Using the power rule:

\[ \log h 3 = 35 \log 3 \]

Question 4

To evaluate \( \log \sqrt{100} 3 \):

First, simplify \( \sqrt{100} \):

\[ \sqrt{100} = 10 \]

Thus,

\[ \log \sqrt{100} 3 = \log_{10} 3 \]

This is the evaluation, but typically in logarithm base 10, this is represented simply as \( \log 3 \).

Question 5

To find the value of \( \log 0.00001 \):

Rewrite \( 0.00001 \) in scientific notation:

\[ 0.00001 = 10^{-5} \]

Using the properties of logarithms:

\[ \log(0.00001) = \log(10^{-5}) = -5 \]

Summary of Answers:

1. \( \log x 4 = 0.5 \log 2 \)
2. Option #3: \( 2 \log 3 \)
3. \( \log h 3 = 35 \log 3 \)
4. \( \log \sqrt{100} 3 = \log 3 \)
5. \( \log 0.00001 = -5 \)