solve the logarithmic equation.
log(5x+7)=1+log(x-9)
log(5x+7) - log(x-9) = 1 , condition: x > 9
log [ (5x+7)/(x-9) ] = 1
(5x+7)/(x-9 = 10^1
10x - 90 = 5x + 7
5x =97
x = 97/5 = 19.4
log (5×+1)=1+log(×-9)
Help
To solve the logarithmic equation log(5x+7) = 1 + log(x-9), we can use the properties of logarithms and algebraic techniques. Here's how you can approach solving this equation step by step:
Step 1: Simplify the equation using the properties of logarithms.
Start by applying the product rule of logarithms, which states that log(a * b) = log(a) + log(b).
log(5x+7) = 1 + log(x-9)
log(5x+7) - log(x-9) = 1
Next, use the quotient rule of logarithms, which states that log(a / b) = log(a) - log(b).
log((5x+7)/(x-9)) = 1
Step 2: Eliminate the logarithm by converting it to an exponential equation.
Remember that logarithms and exponentials are inverse functions. By rewriting the equation in exponential form, we can solve for x.
The logarithmic equation log((5x+7)/(x-9)) = 1 can be rewritten as:
(5x+7)/(x-9) = 10^1
(5x+7)/(x-9) = 10
Step 3: Solve the equation.
To solve for x, we can cross-multiply and simplify the equation.
(5x+7) = 10(x-9)
5x + 7 = 10x - 90
Now, we can isolate the x term on one side of the equation by subtracting 5x from both sides:
7 = 10x - 5x - 90
7 = 5x - 90
Next, add 90 to both sides of the equation:
7 + 90 = 5x - 90 + 90
97 = 5x
Finally, divide both sides by 5 to solve for x:
97/5 = (5x)/5
19.4 = x
Therefore, the solution to the logarithmic equation log(5x+7) = 1 + log(x-9) is x = 19.4.