Questions: Solve the equation. log2(x+5)-log2 x=5 Select the correct choice below and fill in any answer boxes present in your choice. A. x= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.) B. There is no solution.

Solution Steps

To solve the equation \(\log _{2}(x+5)-\log _{2} x=5\), we can use the properties of logarithms. Specifically, we can apply the quotient rule for logarithms, which states that \(\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)\). This allows us to rewrite the equation as a single logarithm. Then, we can convert the logarithmic equation into an exponential equation to solve for \(x\).

1. Use the quotient rule for logarithms to combine the logs: \(\log_2\left(\frac{x+5}{x}\right) = 5\).
2. Convert the logarithmic equation to an exponential equation: \(\frac{x+5}{x} = 2^5\).
3. Solve the resulting equation for \(x\).

We start with the equation: \[ \log_{2}(x+5) - \log_{2}(x) = 5 \] Using the quotient rule for logarithms, we can combine the logs: \[ \log_{2}\left(\frac{x+5}{x}\right) = 5 \]

Next, we convert the logarithmic equation to its exponential form: \[ \frac{x+5}{x} = 2^5 \] This simplifies to: \[ \frac{x+5}{x} = 32 \]

Now, we can cross-multiply to solve for \(x\): \[ x + 5 = 32x \] Rearranging gives: \[ 5 = 32x - x \] \[ 5 = 31x \] Thus, we find: \[ x = \frac{5}{31} \]

Final Answer