To solve the equation [tex]\( 4^{x+3} = 7 \)[/tex] using logarithms, we will proceed step-by-step to isolate [tex]\( x \)[/tex]. The change of base formula for logarithms will be helpful in this process:
1. Apply logarithms to both sides:
We can take the common logarithm (base 10) of both sides of the equation:
[tex]\[
\log(4^{x+3}) = \log(7)
\][/tex]
2. Use the properties of logarithms:
The power rule for logarithms states that [tex]\( \log_b(a^c) = c \cdot \log_b(a) \)[/tex]. Using this property, we can rewrite the left-hand side of the equation:
[tex]\[
(x + 3) \cdot \log(4) = \log(7)
\][/tex]
3. Isolate [tex]\( x + 3 \)[/tex]:
To solve for [tex]\( x + 3 \)[/tex], we need to isolate it by dividing both sides of the equation by [tex]\( \log(4) \)[/tex]:
[tex]\[
x + 3 = \frac{\log(7)}{\log(4)}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Subtract 3 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\log(7)}{\log(4)} - 3
\][/tex]
5. Calculate the values of the logarithms:
Given that:
[tex]\[
\log(7) \approx 0.8450980400142567 \quad \text{and} \quad \log(4) \approx 0.6020599913279623
\][/tex]
6. Perform the division and subtraction:
Using these values, we calculate:
[tex]\[
\frac{0.8450980400142567}{0.6020599913279623} \approx 1.403677461028802
\][/tex]
[tex]\[
x = 1.403677461028802 - 3 \approx -1.596322538971198
\][/tex]
7. Round the result:
Finally, rounding [tex]\( x \)[/tex] to the nearest thousandth, we get:
[tex]\[
x \approx -1.596
\][/tex]
Thus, the solution for [tex]\( x \)[/tex] is approximately [tex]\(\boxed{-1.596}\)[/tex].
