Can someone review my response? Response is below. Thanks

Which property listed above should be used so solve each example? Which property of logarithms should be used write the equation in the proper form before we apply the property?

1. log(x² - 9) - log(x - 3) = log 5

1. log x + log(2x + 1) =1

My response to the question above.

For the equation log(x^2 - 9) - log(x - 3) = log 5, we can use the quotient rule of logarithms to combine the two logarithmic terms on the left-hand side:

log[(x^2 - 9)/(x - 3)] = log 5

To solve for x, we can use Property 1 and set the argument of the logarithm on the left-hand side equal to 5:

(x^2 - 9)/(x - 3) = 5

Then, we can simplify the left-hand side by factoring the numerator as the difference of two squares:

[(x + 3)(x - 3)]/(x - 3) = 5

Canceling the common factor of (x - 3), we obtain:

x + 3 = 5

x = 2

Therefore, the solution to the equation log(x^2 - 9) - log(x - 3) = log 5 is x = 2. In this example, we used the quotient rule of logarithms to combine the logarithmic terms and Property 1 to eliminate the logarithm.

For the equation log x + log(2x + 1) = 1, we can use the product rule of logarithms to combine the two logarithmic terms on the left-hand side:

log[x(2x + 1)] = 1

To solve for x, we can use Property 2 and write the equation in exponential form:

10^1 = x(2x + 1)

10 = 2x^2 + x

Rearranging and setting the equation equal to 0, we obtain:

2x^2 + x - 10 = 0

We can solve this quadratic equation by factoring or using the quadratic formula. The solutions are:

x = -2.5 or x = 2

However, we need to check whether these solutions satisfy the original equation. If x = -2.5, then the argument of one of the logarithmic terms would be negative, which is not allowed, so we reject this solution. Therefore, the only solution to the equation log x + log(2x + 1) = 1 is x = 2. In this example, we used the product rule of logarithms to combine the logarithmic terms and Property 2 to write the equation in exponential form.