Question

1. (u5 l11 p1) select all expressions that are equal to $\log_{2}8$.

a. $\log_{5}20$ b. $\log_{5}125$ c. $\log_{10}100$ d. $\log_{10}1,000$ e. $\log_{3}27$ f. $\log_{10}0.001$
explain why a is not an answer choice.

1. ★(u5 l11 p2) which expression has a greater value: $\log_{10}\frac{1}{100}$ or $\log_{2}\frac{1}{8}$? explain how you know.

Explanation:

Question 8

First, calculate \( \log_{2}8 \):

Step 1: Simplify \( \log_{2}8 \)

We know that \( 2^3 = 8 \), so by the definition of logarithms (\( \log_{a}b = c \) means \( a^c = b \)), \( \log_{2}8 = 3 \).

Now, check each option:

Step 2: Analyze Option a (\( \log_{5}20 \))

We need to see if \( \log_{5}20 = 3 \). By the definition, this would mean \( 5^3 = 20 \). But \( 5^3 = 125
eq 20 \), so \( \log_{5}20
eq 3 \).

Step 3: Analyze Option b (\( \log_{5}125 \))

Check if \( 5^3 = 125 \), which is true. So \( \log_{5}125 = 3 \).

Step 4: Analyze Option c (\( \log_{10}100 \))

Check if \( 10^2 = 100 \), which is true. So \( \log_{10}100 = 2
eq 3 \)? Wait, no, wait: Wait, \( 10^2 = 100 \), so \( \log_{10}100 = 2 \). Wait, but earlier we had \( \log_{2}8 = 3 \). Wait, maybe I made a mistake. Wait, let's recalculate:

Wait, \( \log_{2}8 \): \( 2^3 = 8 \), so that's 3.

Now, \( \log_{5}125 \): \( 5^3 = 125 \), so that's 3.

\( \log_{10}100 \): \( 10^2 = 100 \), so that's 2. Wait, no, maybe I messed up. Wait, \( 10^3 = 1000 \), so \( \log_{10}1000 = 3 \). Ah, right! So option d: \( \log_{10}1000 \). Let's recheck:

Option c: \( \log_{10}100 \): \( 10^2 = 100 \), so \( \log_{10}100 = 2 \).

Option d: \( \log_{10}1000 \): \( 10^3 = 1000 \), so \( \log_{10}1000 = 3 \).

Option e: \( \log_{3}27 \): \( 3^3 = 27 \), so \( \log_{3}27 = 3 \).

Option f: \( \log_{10}0.001 \): \( 10^{-3} = 0.001 \), so \( \log_{10}0.001 = -3 \).

So the expressions equal to \( \log_{2}8 = 3 \) are b, d, e.

Now, explaining why A is not an answer:

To check \( \log_{5}20 \), we use the definition of a logarithm: \( \log_{5}20 = x \) means \( 5^x = 20 \). We know that \( 5^2 = 25 \) and \( 5^1 = 5 \), so \( x \) is between 1 and 2, not 3. Therefore, \( \log_{5}20
eq 3 \), so it's not equal to \( \log_{2}8 \).

Question 9

We need to compare \( \log_{10}\frac{1}{100} \) and \( \log_{2}\frac{1}{8} \).

Step 1: Simplify \( \log_{10}\frac{1}{100} \)

We know that \( \frac{1}{100} = 10^{-2} \), so by the definition of logarithms, \( \log_{10}10^{-2} = -2 \) (since \( \log_{a}a^x = x \)).

Step 2: Simplify \( \log_{2}\frac{1}{8} \)

We know that \( \frac{1}{8} = 2^{-3} \), so by the definition of logarithms, \( \log_{2}2^{-3} = -3 \) (since \( \log_{a}a^x = x \)).

Step 3: Compare the two values

We have \( \log_{10}\frac{1}{100} = -2 \) and \( \log_{2}\frac{1}{8} = -3 \). Since \( -2 > -3 \) (because -2 is to the right of -3 on the number line), \( \log_{10}\frac{1}{100} \) has a greater value.

Final Answers

Question 8

The expressions equal to \( \log_{2}8 \) are:
b. \( \log_{5}125 \), d. \( \log_{10}1000 \), e. \( \log_{3}27 \)

for A: \( \log_{5}20 \) is not 3 because \( 5^x = 20 \) has no integer solution (or \( x \) is not 3, as \( 5^3 = 125
eq 20 \)).

Question 9

\( \log_{10}\frac{1}{100} \) has a greater value.

• \( \log_{10}\frac{1}{100} = \log_{10}10^{-2} = -2 \) (using \( \log_{a}a^x = x \)).
• \( \log_{2}\frac{1}{8} = \log_{2}2^{-3} = -3 \) (using \( \log_{a}a^x = x \)).

Since \( -2 > -3 \), \( \log_{10}\frac{1}{100} \) is greater.

Answer:

• \( \log_{10}\frac{1}{100} = \log_{10}10^{-2} = -2 \) (using \( \log_{a}a^x = x \)).
• \( \log_{2}\frac{1}{8} = \log_{2}2^{-3} = -3 \) (using \( \log_{a}a^x = x \)).

Since \( -2 > -3 \), \( \log_{10}\frac{1}{100} \) is greater.