Questions: Solve for x to the nearest 10th. 100 = 2580(0.94)^x - 180

Transcript text: Solve for $x$ to the nearest 10 th. \[ 100=2580(0.94)^{x}-180 \]

Solution

Solution Steps

Step 1: Isolate the Exponential Term

Start with the original equation: \[ 100 = 2580(0.94)^x - 180 \] Add $ 180 $ to both sides: \[ 100 + 180 = 2580(0.94)^x \] This simplifies to: \[ 280 = 2580(0.94)^x \]

Step 2: Divide by the Coefficient

Next, divide both sides by $ 2580 $: \[ \frac{280}{2580} = (0.94)^x \] This gives us: \[ (0.94)^x = \frac{28}{258} = \frac{14}{129} \]

Step 3: Apply the Logarithm

Take the natural logarithm of both sides: \[ \ln((0.94)^x) = \ln\left(\frac{14}{129}\right) \] Using the property of logarithms, this can be rewritten as: \[ x \cdot \ln(0.94) = \ln\left(\frac{14}{129}\right) \] Now, solve for $ x $: \[ x = \frac{\ln\left(\frac{14}{129}\right)}{\ln(0.94)} \]

Step 4: Round the Result

Finally, round $ x $ to the nearest tenth. The calculated value of $ x $ is approximately $ 35.9 $.