Problem 16 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{7} $$

Step-by-Step Solution

Answer

The expanded logarithmic expression is \(7 \log_b{x}\).

Step 1: Identify the Form

Firstly, identify that the logarithmic expression given, \(\log_b{x^7}\), fits the form \(\log_b{x^n}\), where \(b\) is the base, \(x\) is the argument and \(n\) is the exponent.

Step 2: Apply Logarithm Property

Next, apply the property of logarithms that states \(\log_b{x^n} = n \log_b{x}\). This simplifies the initial expression to \(7 \log_b{x}\).

Step 3: Conclusion

Therefore, the expanded logarithmic expression is \(7 \log_b{x}\). This cannot be further evaluated without specific values for \(b\) and \(x\).