Final answer:
To solve for x in logarithmic equations, follow these steps: isolate the logarithmic expression, apply the property of logarithms, convert the equation into exponential form, solve for x, and check the solution within the domain of the logarithmic function.
Step-by-step explanation:
To solve for x in logarithmic equations, we need to use the properties of logarithms. Here are the steps:
- Isolate the logarithmic expression.
- Apply the property of logarithms to simplify the equation.
- Convert the equation into exponential form.
- Solve for x by isolating x on one side of the equation.
- Check the solution if it is within the domain of the logarithmic function.
Let's look at an example:
Solve for x in the equation log(x + 1) - log(x) = 2.
Step 1: Isolate the logarithmic expression.
log(x + 1) - log(x) = 2
Step 2: Apply the property of logarithms.
log((x + 1)/x) = 2
Step 3: Convert the equation into exponential form.
10^2 = (x + 1)/x
Step 4: Solve for x by isolating x.
100x = x + 1
99x = 1
x = 1/99
Step 5: Check the solution.
Since the domain of the logarithmic function is x > 0, the solution x = 1/99 satisfies the equation.