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Answer :
- Take the natural logarithm of both sides: $\ln(4^{3x}) = \ln(37.9)$.
- Use the power rule of logarithms: $3x \ln(4) = \ln(37.9)$.
- Isolate $x$: $x = \frac{\ln(37.9)}{3 \ln(4)}$.
- Calculate $x$: $x \approx \boxed{0.874}$.
### Explanation
1. Understanding the Problem
We are given the equation $4^{3x} = 37.9$ and asked to find the value of $x$. To solve for $x$, we will use logarithms.
2. Applying Logarithms
Take the natural logarithm of both sides of the equation: $\ln(4^{3x}) = \ln(37.9)$.
3. Simplifying the Equation
Use the power rule of logarithms to simplify the left side: $3x \ln(4) = \ln(37.9)$.
4. Isolating x
Divide both sides by $3 \ln(4)$ to isolate $x$: $x = \frac{\ln(37.9)}{3 \ln(4)}$.
5. Calculating the Value of x
Calculate the value of $x$: $x = \frac{\ln(37.9)}{3 \ln(4)} \approx 0.874$.
6. Final Answer
Therefore, the value of $x$ is approximately $0.874$.
### Examples
Exponential equations like this one are used in various fields, such as calculating the growth of bacteria, determining the decay of radioactive substances, or modeling the growth of investments. For instance, if you invest money in an account with continuously compounded interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving for the time it takes to reach a certain amount involves using logarithms, just like in this problem.
- Use the power rule of logarithms: $3x \ln(4) = \ln(37.9)$.
- Isolate $x$: $x = \frac{\ln(37.9)}{3 \ln(4)}$.
- Calculate $x$: $x \approx \boxed{0.874}$.
### Explanation
1. Understanding the Problem
We are given the equation $4^{3x} = 37.9$ and asked to find the value of $x$. To solve for $x$, we will use logarithms.
2. Applying Logarithms
Take the natural logarithm of both sides of the equation: $\ln(4^{3x}) = \ln(37.9)$.
3. Simplifying the Equation
Use the power rule of logarithms to simplify the left side: $3x \ln(4) = \ln(37.9)$.
4. Isolating x
Divide both sides by $3 \ln(4)$ to isolate $x$: $x = \frac{\ln(37.9)}{3 \ln(4)}$.
5. Calculating the Value of x
Calculate the value of $x$: $x = \frac{\ln(37.9)}{3 \ln(4)} \approx 0.874$.
6. Final Answer
Therefore, the value of $x$ is approximately $0.874$.
### Examples
Exponential equations like this one are used in various fields, such as calculating the growth of bacteria, determining the decay of radioactive substances, or modeling the growth of investments. For instance, if you invest money in an account with continuously compounded interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving for the time it takes to reach a certain amount involves using logarithms, just like in this problem.
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