Solve: 19^(x-5)=3^(x+2)
i know you then go: (x-5) log19= (x+2) log3
i just don't know what goes after?
those logs are just numbers, so
x log19 - 5log19 = x log3 + 2log3
x(log19-log3) = 2log3 + 5log19
x =
2log3 + 5log19
--------------------
log19 - log3
expand:
(x-5) log19= (x+2) log3
xlog19 - 5log19 = xlog3 + 2log3
xlog19 - xlog3 = 2log3 + 5log19
x(log19 - log3) = 2log3 + 5log10
x = (2log3 + 5log19)/(log19 - log3)
= appr 9.1663
To solve the equation 19^(x-5) = 3^(x+2), you are correct that you can take the logarithm of both sides to help simplify the equation:
(x-5) log19 = (x+2) log3
To continue solving the equation, you can distribute the logarithms on both sides:
x log19 - 5 log19 = x log3 + 2 log3
Next, you can gather the terms with x on one side of the equation and the constant terms on the other side:
x log19 - x log3 = 2 log3 + 5 log19
Now, factor out the x terms on the left side:
x(log19 - log3) = 2 log3 + 5 log19
To isolate x, divide both sides of the equation by (log19 - log3):
x = (2 log3 + 5 log19) / (log19 - log3)
This gives you the value of x that solves the equation 19^(x-5) = 3^(x+2).