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Chapter 12: Problem 75

Use the special properties of logarithms to evaluate each expression. \(6^{\log _{6} 9}\)

Short Answer

Expert verified
9

Step by step solution

01

Identify the structure of the expression

The expression given is \(6^{\log_{6} 9}\).
02

Recall the logarithmic identity

Use the property of logarithms: \(a^{\log_{a} b} = b\). This property states that an exponential expression where the base of the exponent matches the base of the logarithm simplifies to the argument of the logarithm.
03

Apply the logarithmic identity

By applying the property \(a^{\log_{a} b} = b\) to the expression \(6^{\log_{6} 9}\), the result directly simplifies to the argument of the logarithm, which in this case is 9.

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Key Concepts

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

logarithmic properties
Logarithmic properties are essential tools in simplifying complex expressions and solving equations. The properties of logarithms include the product rule, quotient rule, and power rule.

For this exercise, we focus on a special logarithmic identity: \(a^{\log_{a} b} = b\)

This identity helps to simplify expressions where the base of the exponential expression matches the base of the logarithm.

By understanding and applying this property, you can transform more complex calculations into simpler ones. It鈥檚 a powerful technique often used in algebra and calculus.
exponential expressions
Exponential expressions involve numbers raised to a power. These expressions have the form \(a^b\), where 'a' is the base and 'b' is the exponent.

In our given exercise \(6^{\log_{6} 9}\), we see an exponential expression. The exponent here is not a simple number but rather a logarithmic term.

When dealing with such expressions, it's useful to recall the relevant logarithmic identities and properties. This allows you to simplify the expression efficiently.

Understanding exponential functions and their relationship to logarithms is key to tackling various mathematical problems.
logarithm simplification
Simplifying logarithmic expressions involves using different properties and identities to rewrite the expression in a simpler form.

In the exercise, we used the property \(a^{\log_{a} b} = b\) to simplify \(6^{\log_{6} 9}\) to 9.

Here are some common strategies for simplifying logarithmic expressions:
  • Combine or split logarithms using properties
  • Convert between exponential and logarithmic forms
  • Use identities to directly simplify


By practicing these strategies, you can become more comfortable with handling logarithmic and exponential expressions, making mathematical problem-solving more straightforward.

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\log _{10}(x+3)+\log _{10}(x-3)$$

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4} 6^{2}$$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{-0.205 x}=9 $$

Solve each equation. \(\log \sqrt{2}(\sqrt{2})^{9}=x\)

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{7}{5}$$
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