Question

lesson 28 - exponential and logarithmic equati
score: 54/100 answered: 10/20
question 11
solve for x:
5^{8x - 4} = 4^{2x - 9}
x =
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Explanation:

Step1: Take natural log on both sides

Take the natural logarithm of both sides of the equation \(5^{8x - 4}=4^{2x - 9}\). Using the property of logarithms \(\ln(a^b)=b\ln(a)\), we get:
\(\ln(5^{8x - 4})=\ln(4^{2x - 9})\)
\((8x - 4)\ln(5)=(2x - 9)\ln(4)\)

Step2: Expand both sides

Expand the left - hand side and the right - hand side:
\(8x\ln(5)-4\ln(5)=2x\ln(4)-9\ln(4)\)

Step3: Move terms with x to one side

Move all terms with \(x\) to the left - hand side and the constant terms to the right - hand side:
\(8x\ln(5)-2x\ln(4)=4\ln(5)-9\ln(4)\)

Step4: Factor out x

Factor out \(x\) from the left - hand side:
\(x(8\ln(5)-2\ln(4)) = 4\ln(5)-9\ln(4)\)

Step5: Solve for x

Divide both sides by \((8\ln(5)-2\ln(4))\) to solve for \(x\):
\(x=\frac{4\ln(5)-9\ln(4)}{8\ln(5)-2\ln(4)}\)
We can also use the change of base formula or calculate the numerical values. Let's calculate the numerical values:
\(\ln(5)\approx1.6094\), \(\ln(4) = 1.3863\)
\(4\ln(5)-9\ln(4)=4\times1.6094 - 9\times1.3863=6.4376-12.4767=-6.0391\)
\(8\ln(5)-2\ln(4)=8\times1.6094-2\times1.3863 = 12.8752 - 2.7726 = 10.1026\)
\(x=\frac{-6.0391}{10.1026}\approx - 0.598\) (or we can keep it in the logarithmic form \(\frac{4\ln5 - 9\ln4}{8\ln5 - 2\ln4}\), and we can also simplify the expression using logarithm properties. For example, \(\ln(4)=\ln(2^2) = 2\ln(2)\), \(\ln(5)\) is just \(\ln(5)\).
\(x=\frac{4\ln5-9\times2\ln2}{8\ln5 - 2\times2\ln2}=\frac{4\ln5 - 18\ln2}{8\ln5-4\ln2}=\frac{2\ln5 - 9\ln2}{4\ln5 - 2\ln2}\))

Answer:

\(x=\frac{4\ln(5)-9\ln(4)}{8\ln(5)-2\ln(4)}\) (or the approximate value \(x\approx - 0.598\))