To solve the exponential equation [tex]\( 6^{3x} = 40 \)[/tex] for [tex]\( x \)[/tex], we need to follow several algebraic steps that make use of logarithms.
Step-by-step solution:
1. Rewrite the equation:
[tex]\[
6^{3x} = 40
\][/tex]
2. Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(6^{3x}) = \ln(40)
\][/tex]
3. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Therefore, we can rewrite the left-hand side:
[tex]\[
3x \cdot \ln(6) = \ln(40)
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( 3 \cdot \ln(6) \)[/tex]:
[tex]\[
x = \frac{\ln(40)}{3 \cdot \ln(6)}
\][/tex]
5. Calculate the natural logarithms:
- The natural logarithm of 6 ([tex]\(\ln(6)\)[/tex]) is approximately 1.791759469228055.
- The natural logarithm of 40 ([tex]\(\ln(40)\)[/tex]) is approximately 3.6888794541139363.
6. Substitute these values into the equation:
[tex]\[
x = \frac{3.6888794541139363}{3 \cdot 1.791759469228055}
\][/tex]
7. Perform the division:
Conduct the division to find [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{3.6888794541139363}{3 \cdot 1.791759469228055} \approx \frac{3.6888794541139363}{5.375278407684165} \approx 0.6862676078025173
\][/tex]
So, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 6^{3x} = 40 \)[/tex] is approximately:
[tex]\[
x \approx 0.6862676078025173
\][/tex]
