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Math  /  Algebra

Question50.3x=5975^{0.3 x}=597
The solution set expressed in terms of logarithms is {ln⁡(597)0.3ln⁡(5)}\left\{\frac{\ln (597)}{0.3 \ln (5)}\right\}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any numb Now use a calculator to obtain a decimal approximation for the solution. The solution set is □\square (Use a comma to separate answers as needed. Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. The equation 50.3x=597 5^{0.3x} = 597 is an exponential equation.
2. We will use logarithms to solve for x x .
3. A calculator will be used to find a decimal approximation.

STEP 2

1. Take the natural logarithm of both sides.
2. Use logarithmic properties to solve for x x .
3. Calculate the decimal approximation using a calculator.

STEP 3

Take the natural logarithm of both sides of the equation:
ln⁡(50.3x)=ln⁡(597) \ln(5^{0.3x}) = \ln(597)

STEP 4

Use the power rule of logarithms, which states that ln⁡(ab)=bln⁡(a)\ln(a^b) = b \ln(a), to bring down the exponent:
0.3x⋅ln⁡(5)=ln⁡(597) 0.3x \cdot \ln(5) = \ln(597)

STEP 5

Solve for x x by dividing both sides by 0.3ln⁡(5) 0.3 \ln(5) :
x=ln⁡(597)0.3ln⁡(5) x = \frac{\ln(597)}{0.3 \ln(5)}

STEP 6

Use a calculator to find the decimal approximation of x x . First, calculate ln⁡(597)\ln(597) and ln⁡(5)\ln(5), then divide:
1. Calculate ln⁡(597)\ln(597).
2. Calculate ln⁡(5)\ln(5).
3. Divide the results:
x≈ln⁡(597)0.3ln⁡(5) x \approx \frac{\ln(597)}{0.3 \ln(5)}
Using a calculator:
ln⁡(597)≈6.389 \ln(597) \approx 6.389 ln⁡(5)≈1.609 \ln(5) \approx 1.609 x≈6.3890.3×1.609 x \approx \frac{6.389}{0.3 \times 1.609} x≈6.3890.4827 x \approx \frac{6.389}{0.4827} x≈13.23 x \approx 13.23
The solution set is:
13.23 \boxed{13.23}

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