Question

lesson 28 - exponential and logarithmic equat
score: 58/100 answered: 11/20
question 12
solve for x:
4^(5x - 8) = 9^(10x - 10)
x =
question help: video message instructor

Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(4^{5x - 8}=9^{10x - 10}\). Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get:
\((5x - 8)\ln(4)=(10x - 10)\ln(9)\)

Step2: Expand both sides

Expand the left - hand side and the right - hand side:
\(5x\ln(4)-8\ln(4)=10x\ln(9)-10\ln(9)\)

Step3: Rearrange terms with x

Move all terms with \(x\) to one side and the constant terms to the other side:
\(5x\ln(4)-10x\ln(9)=8\ln(4)-10\ln(9)\)

Step4: Factor out x

Factor out \(x\) from the left - hand side:
\(x(5\ln(4)-10\ln(9))=8\ln(4)-10\ln(9)\)

Step5: Solve for x

Divide both sides by \((5\ln(4)-10\ln(9))\) to solve for \(x\):

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We can also calculate the numerical values:
\(\ln(4)\approx1.3863\), \(\ln(9)\approx2.1972\)
\(8\ln(4)-10\ln(9)=8\times1.3863 - 10\times2.1972=11.0904-21.972=- 10.8816\)
\(5\ln(4)-10\ln(9)=5\times1.3863-10\times2.1972 = 6.9315 - 21.972=-15.0405\)
\(x=\frac{- 10.8816}{-15.0405}\approx0.7235\)

Answer:

\(x=\frac{8\ln(4)-10\ln(9)}{5\ln(4)-10\ln(9)}\) (or approximately \(x\approx0.72\))