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

Question

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

Answer 1

Let's break down each question one at a time.

Question 1

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

Using the power rule of logarithms, we know that:

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

However, without knowing \( \log x 2 \), we can't solve this directly. We can convert \( \log x = 0.25 \) into an exponential form:

\[ x = 10^{0.25} = \sqrt[4]{10} \quad \text{(assuming base 10)} \]

Now, if we need \( \log x 4 \):

1. Find \( \log x 2 \):

   \( \log x 2 = \log (x) 2 = \frac{\log 2}{\log x} = \frac{\log 2}{0.25} = 4 \log 2 \) (again using base 10)

2. Now compute \( \log x 4 \):

\[ \log x 4 = 2 \cdot 4 \log 2 = 8 \log 2 \]

For a specific numerical evaluation, we need \( \log 2 \):

Using \( \log 2 \approx 0.301 \): \[ 8 \log 2 \approx 8 \times 0.301 \approx 2.408 \]

However, since the question only asks for the expression in log form, we can give it as:

\[ \log x 4 = 8 \log 2 \]

Question 2

A) Which of the following is equivalent to \( \log 9 \)?

1. Option #1: \( \log(3+3) \): \[ \log(6) \quad \text{(not equivalent)} \]

2. Option #2: \( 3 \log 3 \): \[ \log(3^3) = \log(27) \quad \text{(not equivalent)} \]

3. Option #3: \( 2 \log 3 \): \[ \log(3^2) = \log(9) \quad \text{(equivalent)} \]

The answer is: Option #3: \( 2 \log 3 \).

Question 3

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

Using the power rule:

\[ \log h 3 = \frac{\log 3}{\log h} = \frac{\log 3}{35} \]

Since we need to keep it in a fraction form, the answer is:

\[ \log h 3 = \frac{\log 3}{35} \]

Question 4

A) Evaluate \( \log \sqrt{100} 3 \).

First, we find \( \sqrt{100} = 10 \), so we have:

\[ \log_{10} 3 \]

Since we are asked for its numerical value, we calculate:

Using \( \log_{10} 3 \approx 0.477 \): \[ \log \sqrt{100} 3 \approx 0.477 \]

Question 5

A) Find the value of \( \log 0.00001 \).

This can be rewritten as:

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

Now we calculate:

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

So the answer is: \[ \log 0.00001 = -5 \]

Summary of Results

1. \( \log x 4 = 8 \log 2 \)
2. Equivalent to \( \log 9 \) is Option #3: \( 2 \log 3 \)
3. \( \log h 3 = \frac{\log 3}{35} \)
4. \( \log \sqrt{100} 3 \approx 0.477 \)
5. \( \log 0.00001 = -5 \)