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Chapter 6: Problem 15

Expand the logarithmic expression. (See Example 2.) $$\log 10 x^5$$

Short Answer

Expert verified
The expansion of the logarithmic expression \(\log 10x^5\) is \(1 + 5 \log x\)

Step by step solution

01

- Separate terms

Our first step is to separate \(\log 10x^5\) into two expressions. When using the property \(\log(mn) = \log m + \log n\), the given expression \(\log 10x^5\) becomes \(\log 10 + \log x^5\).
02

- Apply log rules

The next step is to apply the property of exponents in logarithmic terminology. This means that \(\log(m^n) = n \log m\), which transforms \(\log x^5\) into \(5 \log x\).
03

- Simplify log 10

The third step to the solution is to simplify \(\log 10\). Given that, in this case, the base of the logarithm is 10, this makes \(\log 10 = 1\).

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Key Concepts

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

Properties of Logarithms
Logarithms have a special set of properties that make calculations simpler and allow us to manipulate expressions easily. Understanding these properties is crucial for expanding and simplifying logarithmic expressions just like \(\log 10x^5\).
  • Product Property: This is one of the most useful properties, stating that the logarithm of a product is the sum of the logs of its factors: \(\log(mn) = \log m + \log n\). This property allows us to break down complex logs into simpler parts.
  • Quotient Property: For division, the log of a quotient becomes the difference of logs: \(\log\left(\frac{m}{n}\right) = \log m - \log n\).
  • Power Property: The log of a power can be simplified by bringing the exponent in front: \(\log(m^n) = n \log m\). This is especially helpful when dealing with terms like \(\log x^5\).
These properties are not just theoretical; they are strategic tools. By decomposing terms using these properties, we can handle logs with greater ease, enabling smoother algebraic manipulation.
Exponent Rules in Logarithms
Exponents and logarithms are deeply interconnected, and the transformation between them is often guided by certain rules. Recognizing these exponent rules within logarithmic contexts helps us transform expressions efficiently.When given an expression like \(\log x^5\), the Power Property comes into play, allowing us to rewrite this as \(5 \log x\). This is because raising a number to a power in a log is equivalent to multiplying the log by that power.
  • Transforming powers to products helps make logarithmic expressions more manageable.
  • Understanding how to maneuver through these transformations is key to simplifying logs in algebraic contexts.
  • This rule is particularly useful when dealing with large exponents, reducing complexity and aiding in calculation.
Through these rules, we can effectively manage and manipulate logarithmic expressions, making them friendlier part of our mathematical toolkit.
Simplification of Logarithms
Simplifying logarithms is all about recognizing when and how to apply the rules effectively. Let's look at how we simplified the expressions step by step.In our given example \(\log 10\), utilizing the knowledge that this log has base 10 allows us to directly simplify \(\log 10\) to \(1\). This is because any log \(\log_b b = 1\). Knowing this can save a lot of computation time.
  • Identify simple logarithms like \(\log 10\) or \(\log 1\), which can be evaluated immediately.
  • Understand the base of the logarithm to simplify expressions accurately.
  • Once you apply the simplifying steps, re-evaluate the expression to ensure no further simplification is possible.
By following these simplification strategies, we turn complex logarithmic expressions into simpler and more manageable forms for further analysis or application.

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Most popular questions from this chapter

The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude \(M\) of the dimmest star that can be seen with a telescope is \(M=5 \log D+2\), where \(D\) is the diameter (in millimeters) of the telescope's objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of \(12 ?\)

In Exercises 13 and 14, describe and correct the error in simplifying the expression. $$ \begin{aligned} \left(4 e^{3 x}\right)^2 &=4 e^{(3 x)(2)} \\ &=4 e^{6 x} \end{aligned} $$

Solve the inequality.\(\ln x \geq 3\)

Solve the equation.\(2^{x+3}=5^{3 x-1}\)

Solve the equation. Check for extraneous solutions. \(\ln x+\ln (x-2)=5\)
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