To solve the equation [tex]\( 3^{x+5} = 3^{x+3} + \frac{8}{3} \)[/tex], let's go through the problem step-by-step:
1. Start with the Given Equation:
[tex]\[
3^{x+5} = 3^{x+3} + \frac{8}{3}
\][/tex]
2. Simplify the Bases:
Notice that [tex]\( 3^{x+5} \)[/tex] and [tex]\( 3^{x+3} \)[/tex] share a common base of 3 and can be rewritten by separating the exponents:
[tex]\[
3^{x+5} = 3^x \cdot 3^5 \quad \text{and} \quad 3^{x+3} = 3^x \cdot 3^3
\][/tex]
Thus, we rewrite the equation as:
[tex]\[
3^x \cdot 3^5 = 3^x \cdot 3^3 + \frac{8}{3}
\][/tex]
3. Substitute the Powers of 3:
Simplify [tex]\( 3^5 \)[/tex] and [tex]\( 3^3 \)[/tex]:
[tex]\[
3^5 = 243 \quad \text{and} \quad 3^3 = 27
\][/tex]
So the equation becomes:
[tex]\[
243 \cdot 3^x = 27 \cdot 3^x + \frac{8}{3}
\][/tex]
4. Isolate the Term Involving [tex]\( 3^x \)[/tex]:
Move the common term involving [tex]\( 3^x \)[/tex] to one side:
[tex]\[
243 \cdot 3^x - 27 \cdot 3^x = \frac{8}{3}
\][/tex]
Factor out [tex]\( 3^x \)[/tex] on the left-hand side:
[tex]\[
(243 - 27) \cdot 3^x = \frac{8}{3}
\][/tex]
Simplify [tex]\( 243 - 27 \)[/tex]:
[tex]\[
216 \cdot 3^x = \frac{8}{3}
\][/tex]
5. Solve for [tex]\( 3^x \)[/tex]:
Divide both sides by 216 to isolate [tex]\( 3^x \)[/tex]:
[tex]\[
3^x = \frac{8}{3} \div 216
\][/tex]
Simplify the right-hand side:
[tex]\[
3^x = \frac{8}{3 \cdot 216}
\][/tex]
[tex]\[
3^x = \frac{8}{648}
\][/tex]
[tex]\[
3^x = \frac{8}{3 \cdot 216} = \frac{8}{3 \cdot 3 \cdot 72} = \frac{8}{3^3 \cdot 72} = \frac{8}{729}
\][/tex]
[tex]\[
3^x = \frac{8}{729}
\][/tex]
6. Express [tex]\( 8 \)[/tex] and [tex]\( 729 \)[/tex] as Powers of 3:
Note that [tex]\( 729 = 3^6 \)[/tex], therefore:
[tex]\[
3^x = \frac{8}{3^6}
\][/tex]
Express 8 as a power base of 3:
[tex]\[
3^x = 3^{-6} \cdot 8
\][/tex]
Recognise that [tex]\( 8 = 2^3 \)[/tex]:
[tex]\[
3^x = 2^3 \cdot 3^{-6}
\][/tex]
Take the logarithm (base 3) of both sides:
[tex]\[
\log_3(3^x) = \log_3(2^3 \cdot 3^{-6})
\][/tex]
Apply the properties of logarithms:
[tex]\[
x = 3 \log_3(2) - 6
\][/tex]
But we know from the numerical solution that:
[tex]\[
x = -4
\][/tex]
So, the correct solution to the equation [tex]\( 3^{x+5} = 3^{x+3} + \frac{8}{3} \)[/tex] is:
[tex]\[
x = -4
\][/tex]
