Answer
$x \approx -2.80$
Work Step by Step
Take the natural logarithm on both sides to obtain
$\ln{(5^{2x+3})}=\ln{(3^{x-1})}.$
Use the power rule $\ln{a^x}=x\ln{a}$ to bring down the exponents:
$(2x+3) \cdot \ln{5} = (x-1) \cdot \ln{3}.$
Use the distributive property to obtain
$2x\ln{5}+3\ln{5}=x\ln{3}-\ln{3}.$
Combine like terms. Put all terms with the variable $x$ on the left side and the rest of the terms on the right side. Note that when terms move from one side to the other, the operation changes to its opposite.
$2x\ln{5} - x\ln{3} = -\ln{3}-3\ln{5}.$
Factor out $x$ on the left side to obtain
$x(2\ln{5}-\ln{3})=-\ln{3}-3\ln{5}.$
Divide both sides by $2\ln{5}-\ln{3}$ to obtain
$x = \dfrac{-\ln{3}-3\ln{5}}{2\ln{5}-\ln{3}}.$
Use a calculator to find the value of $x$. Round-off the answers to two decimal places.
$x \approx -2.80.$
