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Chapter 12: Problem 26

Simplify. $$ \log _{9} 9^{10} $$

Short Answer

Expert verified
The simplified form of \(\log _{9} 9^{10}\text{ is } 10.\)

Step by step solution

01

Recall the Logarithm Power Rule

The logarithm power rule states that \[ \log_b(a^c) = c \log_b(a) \] where \(b\text{, } a\text{ and }c\text{ are positive numbers and } b eq 1.\)
02

Apply the Logarithm Power Rule

In the given expression \( \log_{9} 9^{10} \), let \(a=9\text{, } b=9\text{ and } c=10. \) So let's apply the power rule: \[ \log_{9} 9^{10} = 10 \log_{9}(9) \]
03

Simplify Further

Recall that \(\log_{b}(b) = 1\text{ for any valid base b.} \) So, \[ \log_{9}(9) = 1 \]
04

Final Simplification

Using the result from Step 3 in Step 2, we find: \[10 \log_{9}(9) = 10 \cdot 1 = 10 \]

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Key Concepts

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

logarithm power rule
When you encounter a logarithm expression like \[ \log_b(a^c) \], you should think about the logarithm power rule right away.
This rule states that you can bring the exponent, which is the power on the base inside the logarithm, in front as a multiplier.

In mathematical terms, this appears as follows: \[ \log_b(a^c) = c \log_b(a) \]

This rule simplifies logarithmic expressions significantly, because it transforms an exponentiation inside the logarithm into a multiplication outside it.
It's very useful for breaking down complex logarithmic expressions into simpler parts.
logarithm properties
Logarithms have various properties that can simplify calculations easily. These properties help you handle complex expressions and equations involving logarithms more effectively.

Some key properties include:
  • product rule: \[ \log_b(xy) = \log_b(x) + \log_b(y) \]

  • quotient rule: \[ \log_b\left( \frac{x}{y} \right) = \log_b(x) - \log_b(y) \]

  • change of base formula: \[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \]

  • power rule: \[ \log_b(a^c) = c \log_b(a) \]

One very important property, as used in the original exercise, is that \[ \log_b(b) = 1 \] for any valid base \(b\).

For instance, \[ \log_{9}(9) = 1 \] because any base logarithm of itself is always 1.
basic algebra
Understanding logarithms and their properties often relies on basic algebra concepts. Algebra provides the building blocks for manipulating and simplifying these kinds of mathematical expressions.

Here are some fundamental algebra techniques that are useful:
  • Substitution:
    • Introducing a simpler variable to replace a complex part of an expression can make it easier to manage.

  • Factorization:
    • Breaking down a compound expression into simpler parts that are multiplied together.

  • Distributive Property:
    • Useful for distributing multiplication over addition within parentheses.

    These algebraic strategies are crucial when working with logarithms because they allow more complicated expressions to be simplified step-by-step, just as we saw in the original example.

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Most popular questions from this chapter

Simplify. $$ \left(y^{3}\right)^{5} $$

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Solve. $$ \left|\log _{3} x\right|=2 $$

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A student incorrectly reasons that $$\begin{aligned}\log _{b} \frac{1}{x} &=\log _{b} \frac{x}{x x} \\\&=\log _{b} x-\log _{b} x+\log _{b} x=\log _{b} x\end{aligned}$$ What mistake has the student made?
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