Question
Rewrite the following equation in terms of base e. Express the answer in terms of a natural logarithm, then round to three decimal places.
y equals 1.3 left parenthesis 0.6 right parenthesis Superscript x
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Part 1
Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm.
yequals
enter your response here
y equals 1.3 left parenthesis 0.6 right parenthesis Superscript x
Question content area bottom
Part 1
Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm.
yequals
enter your response here
Answers
Answered by
GPT-4o mini
Given the equation
\[ y = 1.3 (0.6)^x \]
we want to rewrite it in terms of base \( e \).
1. First, we'll isolate \( (0.6)^x \):
\[ (0.6)^x = \frac{y}{1.3} \]
2. Next, we can take the natural logarithm of both sides:
\[ \ln((0.6)^x) = \ln\left(\frac{y}{1.3}\right) \]
3. Using the property of logarithms \( \ln(a^b) = b \ln(a) \), we have:
\[ x \ln(0.6) = \ln\left(\frac{y}{1.3}\right) \]
4. Now, solve for \( x \):
\[ x = \frac{\ln\left(\frac{y}{1.3}\right)}{\ln(0.6)} \]
5. Thus, we can express the original equation in terms of \( y \) and \( e \):
\[ y = 1.3 e^{x \ln(0.6)} \]
So, in the context of the original equation rewritten in terms of \( e \):
**Final Answer:**
\[ y = 1.3 e^{x \ln(0.6)} \]
This represents the equation in the desired form. If you want to round \( \ln(0.6) \) to three decimal places for calculation purposes, you can use:
\[ \ln(0.6) \approx -0.5108 \]
So, rounded to three decimal places:
\[ \ln(0.6) \approx -0.511 \]
Plugging in this value:
\[ y = 1.3 e^{-0.511 x} \]
If you're looking to express the equation specifically using numerical values, please confirm.
\[ y = 1.3 (0.6)^x \]
we want to rewrite it in terms of base \( e \).
1. First, we'll isolate \( (0.6)^x \):
\[ (0.6)^x = \frac{y}{1.3} \]
2. Next, we can take the natural logarithm of both sides:
\[ \ln((0.6)^x) = \ln\left(\frac{y}{1.3}\right) \]
3. Using the property of logarithms \( \ln(a^b) = b \ln(a) \), we have:
\[ x \ln(0.6) = \ln\left(\frac{y}{1.3}\right) \]
4. Now, solve for \( x \):
\[ x = \frac{\ln\left(\frac{y}{1.3}\right)}{\ln(0.6)} \]
5. Thus, we can express the original equation in terms of \( y \) and \( e \):
\[ y = 1.3 e^{x \ln(0.6)} \]
So, in the context of the original equation rewritten in terms of \( e \):
**Final Answer:**
\[ y = 1.3 e^{x \ln(0.6)} \]
This represents the equation in the desired form. If you want to round \( \ln(0.6) \) to three decimal places for calculation purposes, you can use:
\[ \ln(0.6) \approx -0.5108 \]
So, rounded to three decimal places:
\[ \ln(0.6) \approx -0.511 \]
Plugging in this value:
\[ y = 1.3 e^{-0.511 x} \]
If you're looking to express the equation specifically using numerical values, please confirm.