Question

solve. round your answer to the nearest thousandth.
$4^{x - 3} = 77$
$x = \square$

Explanation:

Step1: Take log on both sides

Take the logarithm (we can use common logarithm or natural logarithm, here we use natural logarithm for example) of both sides of the equation \(4^{x - 3}=77\). According to the logarithm property \(\ln a^{b}=b\ln a\), we get \(\ln(4^{x - 3})=\ln(77)\), which simplifies to \((x - 3)\ln(4)=\ln(77)\).

Step2: Solve for x

First, we can express \(x\) from the equation \((x - 3)\ln(4)=\ln(77)\). Divide both sides by \(\ln(4)\): \(x - 3=\frac{\ln(77)}{\ln(4)}\). Then add 3 to both sides: \(x = 3+\frac{\ln(77)}{\ln(4)}\).

Now we calculate the value. We know that \(\ln(77)\approx4.3438\) and \(\ln(4)\approx1.3863\). Then \(\frac{\ln(77)}{\ln(4)}\approx\frac{4.3438}{1.3863}\approx3.133\). Then \(x = 3 + 3.133=6.133\) (rounded to the nearest thousandth).

Answer:

\(x\approx6.133\)