solve the following logarithm equation
log (4x+7)=1+log(x-6)
x=?
please show work
log (4x+7)=1+log(x-6)
log (4x+7) - log(x-6) = 1
log [(4x+7)/(x-6)] = 1
(4x+7)/(x-6) = 10^1 = 10
10x - 60 = 4x + 7
6x = 67
x = 67/6
To solve the logarithmic equation, we need to use logarithmic properties to simplify the equation and then solve for x.
Let's begin by applying the property of logarithms that states log(a) + log(b) = log(ab).
log(4x+7) = 1 + log(x-6)
Now we can combine the logarithms on the right side of the equation using the logarithmic property mentioned above.
log(4x+7) = log(10) + log(x-6)
Since log(10) = 1, we can simplify further:
log(4x+7) = 1 + log(x-6)
Next, we'll use the logarithmic property that states log(a) - log(b) = log(a/b):
log(4x+7) - log(x-6) = 1
Now we can combine the logarithms on the left side of the equation:
log((4x+7)/(x-6)) = 1
To get rid of the logarithm, we can rewrite the equation in exponential form:
(4x+7)/(x-6) = 10^1
Simplifying further:
(4x+7)/(x-6) = 10
To solve for x, we'll cross-multiply:
10(x-6) = 4x+7
Distribute the 10:
10x - 60 = 4x + 7
Combine like terms:
10x - 4x = 7 + 60
6x = 67
Divide both sides by 6:
x = 67/6
Therefore, the solution to the equation log(4x+7) = 1 + log(x-6) is x = 67/6.