To solve the equation [tex]\( (0.3)^{\frac{3x}{5}} = 0.027 \)[/tex], we need to find the value of [tex]\( x \)[/tex]. Let's go through the steps in detail.
1. Take the natural logarithm of both sides:
Taking the natural logarithm (denoted as [tex]\( \ln \)[/tex]) of both sides helps us deal with the exponent more easily.
[tex]\[
\ln \left( (0.3)^{\frac{3x}{5}} \right) = \ln(0.027)
\][/tex]
2. Apply the power rule of logarithms:
The power rule of logarithms states that [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]. Applying this rule, we get:
[tex]\[
\frac{3x}{5} \cdot \ln(0.3) = \ln(0.027)
\][/tex]
3. Isolate the variable [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to isolate it on one side of the equation. We do this by dividing both sides of the equation by [tex]\( \ln(0.3) \)[/tex]:
[tex]\[
\frac{3x}{5} = \frac{\ln(0.027)}{\ln(0.3)}
\][/tex]
Next, multiply both sides by [tex]\(\frac{5}{3}\)[/tex] to completely isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5 \cdot \ln(0.027)}{3 \cdot \ln(0.3)}
\][/tex]
4. Calculate the natural logarithms:
Using the natural logarithms given:
[tex]\[
\ln(0.3) \approx -1.2039728043259361
\][/tex]
[tex]\[
\ln(0.027) \approx -3.611918412977808
\][/tex]
5. Substitute the logarithm values into the equation:
Now substitute the values back into the equation we derived for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5 \cdot (-3.611918412977808)}{3 \cdot (-1.2039728043259361)}
\][/tex]
6. Perform the final calculation:
Simplify the fraction:
[tex]\[
x = \frac{5 \cdot -3.611918412977808}{3 \cdot -1.2039728043259361}
\][/tex]
Notice that the negative signs cancel out:
[tex]\[
x = \frac{5 \cdot 3.611918412977808}{3 \cdot 1.2039728043259361}
\][/tex]
[tex]\[
x = \frac{18.05959206488904}{3.611918412977808}
\][/tex]
[tex]\[
x = 5
\][/tex]
So, the solution to the equation [tex]\( (0.3)^{\frac{3x}{5}} = 0.027 \)[/tex] is [tex]\( x = 5 \)[/tex].
