To solve the exponential equation [tex]\(5^x = 20\)[/tex], we can use logarithms to isolate the variable [tex]\(x\)[/tex]. Here are the detailed steps:
1. Taking the Logarithm of Both Sides:
We start by taking the logarithm of both sides with a common base. For convenience, we'll use the natural logarithm (logarithm base [tex]\(e\)[/tex]):
[tex]\[
\ln(5^x) = \ln(20)
\][/tex]
2. Applying the Power Rule of Logarithms:
Using the logarithm power rule, [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex], we can move the exponent [tex]\(x\)[/tex] in front of the logarithm on the left-hand side of the equation:
[tex]\[
x \cdot \ln(5) = \ln(20)
\][/tex]
3. Isolating [tex]\(x\)[/tex]:
Now, we solve for [tex]\(x\)[/tex] by dividing both sides of the equation by [tex]\(\ln(5)\)[/tex]:
[tex]\[
x = \frac{\ln(20)}{\ln(5)}
\][/tex]
Thus, the exact solution for [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{\ln(20)}{\ln(5)}
\][/tex]
4. Finding the Numerical Value:
To find the numerical value, we can use a calculator to compute the natural logarithms:
- [tex]\(\ln(20) \approx 2.9957\)[/tex]
- [tex]\(\ln(5) \approx 1.6094\)[/tex]
Plugging these values into the equation, we get:
[tex]\[
x \approx \frac{2.9957}{1.6094} \approx 1.8613531161467862
\][/tex]
Therefore, the exact solution for [tex]\(x\)[/tex] is approximately [tex]\(1.8613531161467862\)[/tex].
5. Rounding for Approximation:
Finally, we can round this approximate value to four decimal places:
[tex]\[
x \approx 1.8614
\][/tex]
So, the solutions are:
1) Exact Solution:
[tex]\[
x = \frac{\ln(20)}{\ln(5)} = 1.8613531161467862
\][/tex]
2) Approximation (Rounded to 4 decimals):
[tex]\[
x = 1.8614
\][/tex]
