Certainly! Let's solve the equation step by step:
[tex]\[
4 \cdot 3^{0.2x} = 5
\][/tex]
1. Isolate the exponential term:
Start by dividing both sides of the equation by 4 to isolate the exponential term.
[tex]\[
3^{0.2x} = \frac{5}{4}
\][/tex]
2. Convert to a logarithmic equation:
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (ln) of both sides of the equation.
[tex]\[
\ln(3^{0.2x}) = \ln\left(\frac{5}{4}\right)
\][/tex]
3. Apply the power rule of logarithms:
Use the property of logarithms that [tex]\( \ln(a^b) = b \ln(a) \)[/tex].
[tex]\[
0.2x \cdot \ln(3) = \ln\left(\frac{5}{4}\right)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( 0.2 \cdot \ln(3) \)[/tex].
[tex]\[
x = \frac{\ln\left(\frac{5}{4}\right)}{0.2 \cdot \ln(3)}
\][/tex]
5. Calculate the numerical value:
After substituting the numerical values into the equation, we solve for [tex]\( x \)[/tex].
Therefore, the solution to the equation is:
[tex]\[
x \approx 1.01557006787506
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 4 \cdot 3^{0.2x} = 5 \)[/tex] is approximately [tex]\( 1.01557006787506 \)[/tex].
