Answer:
[tex]x=5[/tex]
Step-by-step explanation:
We have the equation:
[tex]34\cdot 3^{x-2}-2\cdot 3^{x-3}=0.8\cdot 10^{x-2}+10^{x-3}[/tex]
Let's simplify this a bit. Let [tex]u=x-2[/tex]. Then [tex]u-1=x-3[/tex]. We can substitute the exponents:
[tex]34\cdot3^u-2\cdot 3^{u-1}=0.8\cdot 10^u+10^{u-1}[/tex]
Use the properties of exponents, we can write [tex]3^{u-1}=\frac{3^u}{3}[/tex]. We can do the same thing on the right. So:
[tex]34\cdot3^u-2\cdot \frac{3^u}{3}=0.8\cdot 10^u+\frac{10^u}{10}[/tex]
We can now factor out a [tex]3^u[/tex] from the left and a [tex]10^u[/tex] on the right. This yields:
[tex]3^u(34-2\cdot\frac{1}{3})=10^u(0.8+ \frac{1}{10})[/tex]
Evaluate the expressions within the parentheses:
[tex]3^u(34-\frac{2}{3})=10^u(\frac{9}{10})[/tex]
Evaluate:
[tex]3^u(\frac{100}{3})=10^u(\frac{9}{10})[/tex]
Now, let's multiply both sides by [tex]\frac{3}{100}[/tex]. So:
[tex]3^u=10^u(\frac{9}{10})(\frac{3}{100})[/tex]
Also, let's divide both sides by [tex]10^u[/tex]. Multiply on the right:
[tex]\frac{3^u}{10^u}=\frac{27}{1000}[/tex]
Therefore:
[tex]3^u=27\text{ and } 10^u=1000[/tex]
We can now substitute back u. Notice that 27 is the same as 3 cubed and 1000 is the same as 10 cubed. So:
[tex]3^{x-2}=3^3\text{ and } 10^{x-2}=10^3[/tex]
Since they have the same base, their exponents must be equal. Therefore:
[tex]x-2=3[/tex]
Add 2 to both sides:
[tex]x=5[/tex]
So, the value of x is 5.
And we're done!
