Question

solve
$4^{x + 7} = 6$

Explanation:

Step1: Isolate the exponential term

Subtract 7 from both sides of the equation \(4^{x + 7}=6\) to get \(4^{x+7}-7 + 7=6 - 7\)? Wait, no, directly: \(4^{x + 7}=6\). To solve for \(x\), we take the logarithm of both sides. Let's use the natural logarithm (ln) or common logarithm (log). Let's use natural logarithm.
Take \(\ln\) of both sides: \(\ln(4^{x + 7})=\ln(6)\)

Step2: Use logarithm power rule

The power rule of logarithms states that \(\ln(a^{b})=b\ln(a)\). So, \((x + 7)\ln(4)=\ln(6)\)

Step3: Solve for x

First, divide both sides by \(\ln(4)\): \(x + 7=\frac{\ln(6)}{\ln(4)}\)
Then, subtract 7 from both sides: \(x=\frac{\ln(6)}{\ln(4)}-7\)
We can simplify \(\frac{\ln(6)}{\ln(4)}\) as \(\log_{4}(6)\) (by change of base formula \(\log_{a}(b)=\frac{\ln(b)}{\ln(a)}\)). Also, \(\ln(4) = 2\ln(2)\), \(\ln(6)=\ln(2)+\ln(3)\), but we can just compute the numerical value.
Calculate \(\frac{\ln(6)}{\ln(4)}\approx\frac{1.7918}{1.3863}\approx1.292\)
Then \(x\approx1.292 - 7=-5.708\)
Alternatively, using common logarithm: \(\log(4^{x + 7})=\log(6)\)
\((x + 7)\log(4)=\log(6)\)
\(x + 7=\frac{\log(6)}{\log(4)}\approx\frac{0.7782}{0.6021}\approx1.292\)
\(x\approx1.292 - 7=-5.708\)

Answer:

\(x=\frac{\ln(6)}{\ln(4)}-7\) (or approximately \(x\approx - 5.71\))