solve the following logarithm equation
log (4x+7)=1+log(x-6)
x=?
please show work
To solve the logarithm equation, follow these steps:
Step 1: Use the properties of logarithms to simplify the equation.
Start with the equation:
log (4x+7) = 1 + log(x-6)
Use the property of logarithms stating that log(a) + log(b) = log(a*b):
log (4x+7) = log((x-6)*10)
Since the bases of the logarithms on both sides are the same (logarithms with no base specified are usually assumed to be base 10), we can remove the logarithms on both sides:
4x + 7 = (x - 6) * 10
Step 2: Solve the resulting linear equation.
Distribute 10 to the terms inside the parentheses on the right side:
4x + 7 = 10x - 60
Rearrange the equation by bringing like terms to one side:
10x - 4x = 7 + 60
6x = 67
Finally, divide both sides by 6 to solve for x:
x = 67/6
So the solution to the logarithm equation is x = 67/6.