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

Question

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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 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 $\log x4$.

Using the power rule of logarithms, we know that:

$$\log x4=\log x(2^2)=2\log x2$$

However, without knowing $\log x2$, we can't solve this directly. We can convert $\log x=0.25$ into an exponential form:

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

Now, if we need $\log x4$:

1. Find $\log x2$:

   $\log x2=\log (x)2=\frac{\log 2}{\log x}=\frac{\log 2}{0.25}=4\log 2$ (again using base 10)

2. Now compute $\log x4$:

$$\log x4=2\cdot 4\log 2=8\log 2$$

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:

$$\log x4=8\log 2$$

Question 2

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

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

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

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

The answer is: Option #3: $2\log 3$.

Question 3

A) If $\log h=35$, we want to find $\log h3$.

Using the power rule:

$$\log h3=\frac{\log 3}{\log h}=\frac{\log 3}{35}$$

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

$$\log h3=\frac{\log 3}{35}$$

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. $\log x4=8\log 2$
2. Equivalent to $\log 9$ is Option #3: $2\log 3$
3. $\log h3=\frac{\log 3}{35}$
4. $\log \sqrt{100}3\approx 0.477$
5. $\log 0.00001=-5$