- Simplify the equation by dividing by 10 to get \((1.5)^{3x} = 8\).
- Apply the natural logarithm to both sides and use the power rule: \(3x \ln(1.5) = \ln(8)\).
- Solve for \(x\): \(x = \frac{\ln(8)}{3 \ln(1.5)}\).
- Calculate the numerical value and round to the nearest hundredth to get \(x = \boxed{1.71}\).
Explanation
- Isolating the Exponential Term
The given equation is \(10(1.5)^{3x} = 80\). Our goal is to isolate the variable \(x\) and find its value rounded to the nearest hundredth. We first simplify the equation by dividing both sides by 10, which yields:
\[(1.5)^{3x} = \frac{80}{10} = 8.\]
- Applying Logarithms
To solve for the exponent \(3x\), we take the natural logarithm (\(\ln\)) of both sides:
\[\ln((1.5)^{3x}) = \ln(8).\]
Using the power rule for logarithms, which states \(\ln(a^b) = b \cdot \ln(a)\), we can rewrite the left side:
\[3x \cdot \ln(1.5) = \ln(8).\]
- Solving and Rounding the Result
Now, we isolate \(x\) by dividing both sides of the equation by \(3 \cdot \ln(1.5)\):
\[x = \frac{\ln(8)}{3 \cdot \ln(1.5)}.\]
Using numerical values for the logarithms, we calculate:
\[x \approx \frac{2.07944}{3 \times 0.40547} \approx \frac{2.07944}{1.21641} \approx 1.7095.\]
Rounding this result to the nearest hundredth, we get \(x \approx 1.71\).
Examples
Exponential equations appear in many real-life situations, such as calculating compound interest, modeling population growth, or even determining the decay of radioactive materials. For instance, if an investment grows according to a specific interest rate, you might need to solve an exponential equation to determine how many years it will take to reach a target amount. Understanding how to use logarithms to solve these problems is a fundamental skill in finance and science.