To solve the equation [tex]\( 4^{x+3} = 7^{x-6} \)[/tex], we'll use logarithms to simplify and solve for [tex]\( x \)[/tex]. Here are the steps:
1. Take the natural logarithm (ln) of both sides of the equation:
[tex]\[
\ln(4^{x+3}) = \ln(7^{x-6})
\][/tex]
2. Use the properties of logarithms to bring the exponents down as coefficients:
[tex]\[
(x + 3) \cdot \ln(4) = (x - 6) \cdot \ln(7)
\][/tex]
3. Distribute the natural logarithms:
[tex]\[
x \cdot \ln(4) + 3 \cdot \ln(4) = x \cdot \ln(7) - 6 \cdot \ln(7)
\][/tex]
4. Group all terms involving [tex]\( x \)[/tex] on one side of the equation and constant terms on the other side:
[tex]\[
x \cdot \ln(4) - x \cdot \ln(7) = -6 \cdot \ln(7) - 3 \cdot \ln(4)
\][/tex]
5. Factor [tex]\( x \)[/tex] out of the left side of the equation:
[tex]\[
x \cdot (\ln(4) - \ln(7)) = -6 \cdot \ln(7) - 3 \cdot \ln(4)
\][/tex]
6. Isolate [tex]\( x \)[/tex] by dividing both sides by [tex]\( (\ln(4) - \ln(7)) \)[/tex]:
[tex]\[
x = \frac{-6 \cdot \ln(7) - 3 \cdot \ln(4)}{\ln(4) - \ln(7)}
\][/tex]
Next, we need to find the values of the natural logarithms:
Let:
[tex]\[
\ln(4) \approx 1.386294
\][/tex]
[tex]\[
\ln(7) \approx 1.945910
\][/tex]
Substitute these values into the equation:
[tex]\[
x = \frac{-6 \cdot 1.945910 - 3 \cdot 1.386294}{1.386294 - 1.945910}
\][/tex]
Calculate the numerator and denominator separately:
Numerator:
[tex]\[
-6 \cdot 1.945910 \approx -11.675460
\][/tex]
[tex]\[
-3 \cdot 1.386294 \approx -4.158883
\][/tex]
[tex]\[
-11.675460 - 4.158883 \approx -15.834343
\][/tex]
Denominator:
[tex]\[
1.386294 - 1.945910 \approx -0.559616
\][/tex]
Now, divide these values to find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-15.834343}{-0.559616} \approx 28.295
\][/tex]
Therefore, the solution set for the equation [tex]\( 4^{x+3} = 7^{x-6} \)[/tex] rounded to three decimal places is:
[tex]\[
\boxed{28.295}
\][/tex]
