Let's solve the equation [tex]\( 2^{x-3} = 3^{2x} \)[/tex] step by step.
### Step 1: Rewrite the equation using logarithms
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (or log base 10) of both sides of the equation. However, let's use the natural logarithm [tex]\( \ln \)[/tex] for simplicity.
[tex]\[ \ln(2^{x-3}) = \ln(3^{2x}) \][/tex]
### Step 2: Simplify using logarithm properties
Using the properties of logarithms, specifically [tex]\( \ln(a^b) = b \ln(a) \)[/tex], we simplify both sides:
[tex]\[ (x - 3) \ln(2) = 2x \ln(3) \][/tex]
### Step 3: Distribute the logarithms
Distribute [tex]\( \ln(2) \)[/tex] and [tex]\( \ln(3) \)[/tex] across their respective terms:
[tex]\[ x \ln(2) - 3 \ln(2) = 2x \ln(3) \][/tex]
### Step 4: Collect all terms involving [tex]\( x \)[/tex] on one side
To isolate [tex]\( x \)[/tex], move all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[ x \ln(2) - 2x \ln(3) = 3 \ln(2) \][/tex]
Factor [tex]\( x \)[/tex] out of the left-hand side:
[tex]\[ x (\ln(2) - 2 \ln(3)) = 3 \ln(2) \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Now, solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( (\ln(2) - 2 \ln(3)) \)[/tex]:
[tex]\[ x = \frac{3 \ln(2)}{\ln(2) - 2 \ln(3)} \][/tex]
This expression can be simplified further, but we have the general form of [tex]\( x \)[/tex].
However, the exact simplified answer given previously, using different forms of logarithm properties and expressions, is represented as:
[tex]\[ x = \log \left(2^{\frac{3}{\log(\frac{2}{9})}}\right) \][/tex]
Both methods lead us towards a complex form involving logarithms, and while it might seem confusing, it represents the solution to the equation [tex]\( 2^{x-3} = 3^{2x} \)[/tex]. Thus, the final form of [tex]\( x \)[/tex] involves logarithmic expressions.
