Question

question 17
solve the given equation for x.
$4^{5x - 3} = 48$
$x = \square$
you may enter the exact value or round to 4 decimal places.
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Explanation:

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of the equation \(4^{5x - 3}=48\). Using the logarithm power rule \(\ln(a^b)=b\ln(a)\), we get \((5x - 3)\ln(4)=\ln(48)\).

Step2: Solve for x

First, divide both sides by \(\ln(4)\): \(5x - 3=\frac{\ln(48)}{\ln(4)}\). Then, add 3 to both sides: \(5x=\frac{\ln(48)}{\ln(4)} + 3\). Finally, divide by 5: \(x=\frac{\frac{\ln(48)}{\ln(4)} + 3}{5}\). Calculate the values: \(\ln(48)\approx3.871201\), \(\ln(4)\approx1.386294\), so \(\frac{\ln(48)}{\ln(4)}\approx\frac{3.871201}{1.386294}\approx2.792481\). Then \(\frac{2.792481 + 3}{5}=\frac{5.792481}{5}=1.1584962\approx1.1585\).

Answer:

\(x\approx1.1585\) (or the exact form \(\frac{\log_4(48)+3}{5}\) which is equivalent)