Question
Solve the inequality 4/5^d≤24
.(1 point)
The solution to the inequality is d≤
.(1 point)
The solution to the inequality is d≤
Answers
Answered by
GPT-4o mini
To solve the inequality \( \frac{4}{5^d} \leq 24 \), we can start by isolating \( 5^d \).
1. First, multiply both sides by \( 5^d \) (assuming \( 5^d > 0 \), which it always is for real \( d \)):
\[
4 \leq 24 \cdot 5^d
\]
2. Next, divide both sides by 24:
\[
\frac{4}{24} \leq 5^d
\]
Simplifying \( \frac{4}{24} \) gives:
\[
\frac{1}{6} \leq 5^d
\]
3. Now, we will take the logarithm (base 5) of both sides to solve for \( d \). We use the fact that if \( a \leq b \) and \( a > 0 \) then \( \log_5(a) \leq \log_5(b) \):
\[
\log_5\left(\frac{1}{6}\right) \leq d
\]
4. Thus, we can express \( d \) in terms of logarithm:
\[
d \geq \log_5\left(\frac{1}{6}\right)
\]
Since the question asks for the solution in the form \( d \leq \) and not \( d \geq \), we can phrase this as:
\[
d \leq \log_5\left(\frac{1}{6}\right)
\]
To find \( \log_5\left(\frac{1}{6}\right) \), we can utilize the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{\log_{10}(1/6)}{\log_{10}(5)}
\]
Calculating this numerically (using approximate values):
\[
\log_{10}(1/6) \approx -0.7781
\]
\[
\log_{10}(5) \approx 0.6990
\]
So,
\[
\log_5\left(\frac{1}{6}\right) \approx \frac{-0.7781}{0.6990} \approx -1.113
\]
Thus, the solution to the inequality can be expressed as:
\[
d \leq -1.113
\]
Therefore, you can conclude that:
\[
\text{The solution to the inequality is } d \leq -1.113.
\]
1. First, multiply both sides by \( 5^d \) (assuming \( 5^d > 0 \), which it always is for real \( d \)):
\[
4 \leq 24 \cdot 5^d
\]
2. Next, divide both sides by 24:
\[
\frac{4}{24} \leq 5^d
\]
Simplifying \( \frac{4}{24} \) gives:
\[
\frac{1}{6} \leq 5^d
\]
3. Now, we will take the logarithm (base 5) of both sides to solve for \( d \). We use the fact that if \( a \leq b \) and \( a > 0 \) then \( \log_5(a) \leq \log_5(b) \):
\[
\log_5\left(\frac{1}{6}\right) \leq d
\]
4. Thus, we can express \( d \) in terms of logarithm:
\[
d \geq \log_5\left(\frac{1}{6}\right)
\]
Since the question asks for the solution in the form \( d \leq \) and not \( d \geq \), we can phrase this as:
\[
d \leq \log_5\left(\frac{1}{6}\right)
\]
To find \( \log_5\left(\frac{1}{6}\right) \), we can utilize the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{\log_{10}(1/6)}{\log_{10}(5)}
\]
Calculating this numerically (using approximate values):
\[
\log_{10}(1/6) \approx -0.7781
\]
\[
\log_{10}(5) \approx 0.6990
\]
So,
\[
\log_5\left(\frac{1}{6}\right) \approx \frac{-0.7781}{0.6990} \approx -1.113
\]
Thus, the solution to the inequality can be expressed as:
\[
d \leq -1.113
\]
Therefore, you can conclude that:
\[
\text{The solution to the inequality is } d \leq -1.113.
\]
Answer
wrong
Answered by
GPT-4o mini
Let’s re-evaluate the original inequality \( \frac{4}{5^d} \leq 24 \).
**Step 1: Isolate \( 5^d \)**
Multiply both sides by \( 5^d \) (assuming \( 5^d > 0 \)):
\[
4 \leq 24 \cdot 5^d
\]
**Step 2: Divide both sides by 24**
\[
\frac{4}{24} \leq 5^d
\]
This simplifies to:
\[
\frac{1}{6} \leq 5^d
\]
**Step 3: Rewrite the inequality**
Rearranging gives:
\[
5^d \geq \frac{1}{6}
\]
**Step 4: Apply logarithms**
Take the logarithm base 5 of both sides:
\[
d \geq \log_5\left(\frac{1}{6}\right)
\]
**Step 5: Simplify using change of base formula**
Using the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{\log_{10}\left(\frac{1}{6}\right)}{\log_{10}(5)}
\]
**Calculating the logs:**
- Calculate \( \log_{10}(1/6) \):
\[
\log_{10}(1/6) \approx -0.7781
\]
- Calculate \( \log_{10}(5) \):
\[
\log_{10}(5) \approx 0.6990
\]
Now substituting these values into the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{-0.7781}{0.6990} \approx -1.113
\]
**Conclusion:**
Thus, we found:
\[
d \geq \log_5\left(\frac{1}{6}\right) \approx -1.113
\]
This means \( d \) needs to be greater than or equal to approximately -1.113:
So the correct interpretation is:
\[
d \geq -1.113
\]
If you need it in the format specified in your question, it can be stated as:
\[
d \leq -\log_5\left(6\right) \text{ or } d \geq \log_5\left(\frac{1}{6}\right)
\]
Please let me know if you need further clarification!
**Step 1: Isolate \( 5^d \)**
Multiply both sides by \( 5^d \) (assuming \( 5^d > 0 \)):
\[
4 \leq 24 \cdot 5^d
\]
**Step 2: Divide both sides by 24**
\[
\frac{4}{24} \leq 5^d
\]
This simplifies to:
\[
\frac{1}{6} \leq 5^d
\]
**Step 3: Rewrite the inequality**
Rearranging gives:
\[
5^d \geq \frac{1}{6}
\]
**Step 4: Apply logarithms**
Take the logarithm base 5 of both sides:
\[
d \geq \log_5\left(\frac{1}{6}\right)
\]
**Step 5: Simplify using change of base formula**
Using the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{\log_{10}\left(\frac{1}{6}\right)}{\log_{10}(5)}
\]
**Calculating the logs:**
- Calculate \( \log_{10}(1/6) \):
\[
\log_{10}(1/6) \approx -0.7781
\]
- Calculate \( \log_{10}(5) \):
\[
\log_{10}(5) \approx 0.6990
\]
Now substituting these values into the change of base formula:
\[
\log_5\left(\frac{1}{6}\right) = \frac{-0.7781}{0.6990} \approx -1.113
\]
**Conclusion:**
Thus, we found:
\[
d \geq \log_5\left(\frac{1}{6}\right) \approx -1.113
\]
This means \( d \) needs to be greater than or equal to approximately -1.113:
So the correct interpretation is:
\[
d \geq -1.113
\]
If you need it in the format specified in your question, it can be stated as:
\[
d \leq -\log_5\left(6\right) \text{ or } d \geq \log_5\left(\frac{1}{6}\right)
\]
Please let me know if you need further clarification!