\( \log _ { b } x ^ { r } = r \log _ { b } x \)

The equation \( \log_{b} x^{r} = r \log_{b} x \) is a property of logarithms. This property states that the logarithm of a power can be expressed as the exponent multiplied by the logarithm of the base. Let's break it down step by step: 1. **Understanding the Left Side**: - The left side of the equation is \( \log_{b} x^{r} \). This means we are taking the logarithm of \( x^{r} \) with base \( b \). 2. **Using the Power Rule of Logarithms**: - The power rule states that \( \log_{b} (a^{c}) = c \cdot \log_{b} a \). - Applying this rule to the left side, we get: \[ \log_{b} x^{r} = r \cdot \log_{b} x \] 3. **Comparing Both Sides**: - Now, we can see that both sides of the equation are equal: \[ r \cdot \log_{b} x = r \cdot \log_{b} x \] Thus, the equation \( \log_{b} x^{r} = r \log_{b} x \) is indeed true and is a direct application of the power rule of logarithms. If you have any specific questions or need further clarification on this topic, feel free to ask!