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Chapter 4: Problem 87

Describe the power rule for logarithms and give an example.

Short Answer

Expert verified
The power rule for logarithms states that the exponent within the logarithm can be treated as a coefficient: \( log_b \, (a^x) = x \cdot log_b \, a \). An instance of this rule: \( log_2 (8^3) = 3 \cdot log_2 8 \), which eventually simplifies to 9.

Step by step solution

01

Defining Power Rule for Logarithms

The power rule is a basic theorem in logarithms that states that for any number \(x\) and any positive integer \(a\), \(log_b \, (a^x) = x \cdot log_b \, a\). This rule is very useful when simplifying logarithmic expressions.
02

Creating and solving an example

Let's apply the power rule of logarithms to the following example: Calculate \( log_2 (8^3) \).Using the power rule, the expression can be rewritten as: \( log_2 (8^3) = 3 \cdot log_2 8 \). Since \( 2^3 = 8 \), \( log_2 8 = 3 \). So,\( 3 \cdot log_2 8 = 3 \cdot 3 = 9 \).
03

Interpreting Results

From the solution, it can be inferred that the power rule is a straightforward way to manage powers inside a logarithm. The presented solution also shows that applying this rule, the expression simplifies to a simple multiplication.

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Most popular questions from this chapter

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (3 x+7)-\log x $$

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3}(x-2) $$

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. $$ \log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right] $$
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