Simplify 3logb^x + 1/2logb(x+1) to its simplest form

Answer 1

To simplify the expression \( 3 \log_b x + \frac{1}{2} \log_b (x+1) \), we can use properties of logarithms.

1. Use the power rule: \( a \log_b c = \log_b (c^a) \).

   For the first term: \[ 3 \log_b x = \log_b (x^3) \] For the second term: \[ \frac{1}{2} \log_b (x+1) = \log_b ((x+1)^{1/2}) = \log_b \sqrt{x+1} \]

2. Combine the logarithms: We can now combine these two logarithmic expressions. The property we use here is \( \log_b a + \log_b c = \log_b (a \cdot c) \):

   \[ 3 \log_b x + \frac{1}{2} \log_b (x+1) = \log_b (x^3) + \log_b (\sqrt{x+1}) = \log_b (x^3 \cdot \sqrt{x+1}) \]

3. Final expression: Thus, the expression simplifies to:

   \[ \log_b (x^3 \sqrt{x+1}) \]

This is the simplest form of the original expression.