Question

assignment 6.3: exponential equations and ine
score: 9/20 answered: 9/20
question 10
find the solution of the exponential equation
15e^{x}-11 = 12
in terms of logarithms, or correct to four decimal places.
x=
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Explanation:

Step1: Isolate the exponential term

Add \(11\) to both sides of the equation \(15e^{x}-11 = 12\).

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Then divide both sides by \(15\): \(e^{x}=\frac{23}{15}\)

Step2: Take the natural logarithm of both sides

Since \(y = e^{x}\) and \(x=\ln(y)\) (by the property of the natural logarithm, \(\ln(e^{x})=x\) for all real \(x\)), when \(e^{x}=\frac{23}{15}\), we have \(x=\ln(\frac{23}{15})\)

Step3: Calculate the numerical value

Using a calculator, \(\ln(\frac{23}{15})=\ln(23)-\ln(15)\approx3.1355 - 2.7080=0.4275\)

Answer:

\(x = 0.4275\)