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Chapter 12: Problem 22

Express as an equivalent expression that is a product. $$\log _{c} M^{-5}$$

Short Answer

Expert verified
-5 log_c M

Step by step solution

01

Recall the logarithm power rule

The logarithm power rule states that \(\text{log}_a(x^b) = b \text{log}_a(x)\). This rule will help in transforming the given logarithm expression.
02

Apply the logarithm power rule to the given expression

Use the rule on \(\text{log}_c M^{-5}\), giving \(-5 \text{log}_c M\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Power Rule
The logarithm power rule is a fundamental property of logarithms that simplifies expressions where a logarithm contains an exponent. The rule states that \[ \text{log}_a(x^b) = b \text{log}_a(x) \].
This means you can move the exponent inside the logarithm to the front as a coefficient.
For example: \( \text{log}_2(8^3) \) can be written as \( 3 \text{log}_2(8) \).
In the given exercise, we have \( \text{log}_c(M^{-5}) \), so we apply the power rule:
\( \text{log}_c(M^{-5}) = -5 \text{log}_c(M) \).
This converts the expression into a product, making it easier to handle in further calculations.
Logarithmic Expressions
Logarithmic expressions often involve transforming and simplifying logs to a more usable form.
The logarithm power rule is essential in this transformation. Let's break it down:
  • A logarithm expression like \( \text{log}(x^y) \) shows that the variable inside the log is raised to a power.
  • Applying the power rule, \( \text{log}(x^y) = y \text{log}(x) \), simplifies the expression by bringing the exponent y outside as a coefficient.
In our example, \( \text{log}_c(M^{-5}) \) simplifies to \( -5 \text{log}_c(M) \).
By rewriting logarithmic expressions, we can reduce complexity and improve clarity in mathematics.
Negative Exponents
Negative exponents are another key concept that can influence logarithmic expressions.
When encountering a negative exponent, it can be rewritten to indicate a reciprocal.
For instance, \( x^{-n} = \frac{1}{x^n} \).
This is important when dealing with logarithms, as seen in our exercise:
  • Given \( M^{-5} \), you recognize the negative exponent.
  • Applying the logarithm power rule to \( \text{log}_c(M^{-5}) \) results in \( -5 \text{log}_c(M) \), effectively handling the negative exponent and simplifying the expression.
Understanding negative exponents helps in manipulating expressions to make them more straightforward.
Remember that the base of the logarithm remains unchanged during these transformations.

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Most popular questions from this chapter

Solve. $$ \log _{4}(3 x-2)=2 $$

Explain why we say that "a logarithm is an exponent"

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Express as an equivalent expression that is a sum or a difference of logarithms and, if possible, simplify. If \(\log _{a} x=2, \log _{a} y=3,\) and \(\log _{a} z=4,\) what is $$\log _{a} \frac{\sqrt[3]{x^{2} z}}{\sqrt[3]{y^{2} z^{-2}}} ?$$

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{1}{3}$$
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