To solve the exponential equation [tex]\(2^{x-3} = 3\)[/tex], follow these steps:
1. Rewrite the equation: Notice that [tex]\(2^{x-3}\)[/tex] is an exponential expression.
[tex]\[
2^{x-3} = 3
\][/tex]
2. Apply logarithms: To isolate [tex]\(x\)[/tex], take the logarithm of both sides of the equation. Using the natural logarithm [tex]\( \ln \)[/tex]:
[tex]\[
\ln(2^{x-3}) = \ln(3)
\][/tex]
3. Use the power rule of logarithms: The power rule states that [tex]\(\ln(a^b) = b \ln(a)\)[/tex]. Applying this rule:
[tex]\[
(x-3) \ln(2) = \ln(3)
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Isolate [tex]\(x\)[/tex] by dividing both sides by [tex]\(\ln(2)\)[/tex]:
[tex]\[
x-3 = \frac{\ln(3)}{\ln(2)}
\][/tex]
5. Add 3 to both sides: To isolate [tex]\(x\)[/tex], add 3 to both sides:
[tex]\[
x = 3 + \frac{\ln(3)}{\ln(2)}
\][/tex]
So, the exact solution is:
[tex]\[
x = 3 + \frac{\ln(3)}{\ln(2)} \text{, which simplifies to } \frac{\ln(24)}{\ln(2)}
\][/tex]
1. Exact solution: [tex]\(\boxed{\frac{\ln(24)}{\ln(2)}}\)[/tex]
2. Approximate the value: To approximate the value of [tex]\(x\)[/tex], we can calculate the numerical value of the expression:
[tex]\[
x \approx 4.585
\][/tex]
So, the approximation to 4 decimal places is:
[tex]\[
\boxed{4.585}
\][/tex]
