QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places.
Studdy Solution
STEP 1
1. The equation is an exponential equation.
2. We will use logarithms to solve for since the bases are not easily comparable.
STEP 2
1. Take the logarithm of both sides.
2. Use the properties of logarithms to simplify.
3. Solve for .
4. Provide both the exact expression and a decimal approximation.
STEP 3
Take the logarithm of both sides of the equation. You can use either the natural logarithm () or the common logarithm (). Here, we will use the natural logarithm:
STEP 4
Apply the power rule of logarithms, which states that , to bring the exponent down:
STEP 5
Solve for . First, divide both sides by :
Next, add 8 to both sides:
Finally, divide by 2 to solve for :
STEP 6
Calculate the decimal approximation. Use a calculator to find the values of the logarithms and then compute :
The exact expression for is:
The decimal approximation for is:
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