To solve for [tex]\( x \)[/tex] in the equation [tex]\( 5^x = 20 \)[/tex], we can use logarithms. Here is the detailed, step-by-step solution:
1. Write the equation in logarithmic form:
The equation is [tex]\( 5^x = 20 \)[/tex].
2. Apply the logarithm to both sides:
We can take the natural logarithm (or any logarithm, but we'll use the natural logarithm for this explanation) of both sides to help us solve for [tex]\( x \)[/tex]:
[tex]\[
\ln(5^x) = \ln(20)
\][/tex]
3. Use the properties of logarithms:
One useful property of logarithms is that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Applying this property, we get:
[tex]\[
x \cdot \ln(5) = \ln(20)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Now, we can isolate [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\(\ln(5)\)[/tex]:
[tex]\[
x = \frac{\ln(20)}{\ln(5)}
\][/tex]
5. Evaluate the expression:
Using the natural logarithm values for 20 and 5, which are [tex]\(\ln(20) \approx 2.9957\)[/tex] and [tex]\(\ln(5) \approx 1.6094\)[/tex], respectively, we can calculate:
[tex]\[
x \approx \frac{2.9957}{1.6094} \approx 1.8613531161467862
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies [tex]\( 5^x = 20 \)[/tex] is approximately [tex]\( 1.8613531161467862 \)[/tex].
