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Chapter 4: Problem 16

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. $$ \log _{b} x^{7} $$

Short Answer

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

Step by step solution

01

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.
02

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 \(7 \log_b{x}\).
03

Conclusion

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

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