Problem 37
Question
Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7^{x+2}=410$$
Step-by-Step Solution
Verified Answer
The solution is \(x \approx 3.37\).
1Step 1: Rewrite the equation
Write the equation in your preferred form. The given equation is \(7^{x+2}=410\).
2Step 2: Take the natural logarithm of both sides
Due to the properties of logarithms, this step allows us to ‘move’ the exponent (x+2) in front of the logarithm: \(\ln(7^{x+2}) = \ln(410)\). This simplifies to: \((x+2)*\ln(7) = \ln(410)\).
3Step 3: Solve for x
Isolate x by first subtracting 2 from both sides and then divide by \(\ln(7)\): \(x = \frac{\ln(410)}{\ln(7)} - 2\).
4Step 4: Simplify and Approximate
Using a calculator, compute the decimal approximation of x, correct to two decimal places: \(x \approx 3.37\).
Other exercises in this chapter
Problem 37
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