Certainly! Let's solve the equation:
[tex]\[ 5^{3 + t} = 6^{2x - 1} \][/tex]
1. Take the natural logarithm of both sides:
[tex]\[ \ln(5^{3 + t}) = \ln(6^{2x - 1}) \][/tex]
2. Use the properties of logarithms to bring the exponents down:
[tex]\[ (3 + t) \cdot \ln(5) = (2x - 1) \cdot \ln(6) \][/tex]
3. Isolate the term involving [tex]\( t \)[/tex]:
[tex]\[ 3 \cdot \ln(5) + t \cdot \ln(5) = 2x \cdot \ln(6) - \ln(6) \][/tex]
4. Simplify and solve for [tex]\( t \)[/tex]:
[tex]\[ t \cdot \ln(5) = 2x \cdot \ln(6) - \ln(6) - 3 \cdot \ln(5) \][/tex]
[tex]\[ t \cdot \ln(5) = \ln(6) (2x - 1) - 3 \cdot \ln(5) \][/tex]
[tex]\[ t = \frac{\ln(6) (2x - 1) - 3 \cdot \ln(5)}{\ln(5)} \][/tex]
[tex]\[ t = \frac{2x \cdot \ln(6) - \ln(6) - 3 \cdot \ln(5)}{\ln(5)} \][/tex]
5. Combine like terms in the numerator:
[tex]\[ t = \frac{2x \cdot \ln(6) - \ln(6) - 3 \cdot \ln(5)}{\ln(5)} \][/tex]
[tex]\[ t = \frac{2x \cdot \ln(6) - \ln(6) - 3 \cdot \ln(5)}{\ln(5)} \][/tex]
[tex]\[ t = \frac{2x \cdot \ln(6) - \ln(6) - 3 \cdot \ln(5)}{\ln(5)} \][/tex]
[tex]\[ t = \frac{2x \cdot \ln(6) - (\ln(6) + 3 \cdot \ln(5))}{\ln(5)} \][/tex]
6. Recognize that [tex]\(\ln(6) = \ln(2 \times 3) = \ln(2) + \ln(3)\)[/tex] and [tex]\(\ln(36) = \ln(6^2) = 2 \ln(6)\)[/tex]:
[tex]\[ t = \frac{\ln(36) \cdot x - (\ln(6) + 3\ln(5))}{\ln(5)} \][/tex]
Thus, the solution to the equation [tex]\( 5^{3 + t} = 6^{2x - 1} \)[/tex] is:
[tex]\[ t = \frac{x \cdot \ln(36) - \ln(750)}{\ln(5)} \][/tex]
where [tex]\(\ln(750) = \ln(5^3 \cdot 6) = 3\ln(5) + \ln(6)\)[/tex].
