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Chapter 10: Problem 167

In the following exercises, find the exact value of each logarithm without using a calculator. $$ \log _{6} 36 $$

Short Answer

Expert verified
\log_6 36 = 2

Step by step solution

01

Recall the Property of Logarithms

The logarithm \( \log_b a \) asks for the power to which the base \( b \) must be raised to get \( a \). In this case, you need to find the exponent \( x \) such that \( 6^x = 36 \).
02

Express 36 as a Power of 6

Recognize that 36 can be written as a power of 6. Notice that \( 36 = 6^2 \). So, \( 6^x = 6^2 \).
03

Equate the Exponents

Since the bases are the same, you can equate the exponents from the equation \( 6^x = 6^2 \). Therefore, \( x = 2 \).
04

Write the Final Answer

Now you can say that \( \log_6 36 = 2 \).

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Key Concepts

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

Properties of Logarithms
Logarithms help us understand exponents in a different way. The logarithm \( \log_b a \) asks the question: 'What power must we raise the base \( b \) to, to get \( a \)?'
There are important properties that make logarithms very useful:
  • Product Property: \( \log_b (xy) = \log_b x + \log_b y \) - This means multiplying numbers inside the log turns into addition outside.
  • Quotient Property: \( \log_b \left( \frac{x}{y}\right) = \log_b x - \log_b y \) - Dividing numbers inside the log turns into subtraction outside.
  • Power Property: \( \log_b (x^y) = y \log_b x \) - Raising a number inside the log to a power turns into multiplication outside.
Understanding these properties allows us to simplify logarithmic expressions and solve complex problems with more ease.
Expressing Numbers as Powers
To find the value of a logarithm, it's often helpful to express the number inside the log as a power of the base. For example, the logarithm \( \log_6 36 \) means: 'To what power should 6 be raised to get 36?'
To solve this, we need to recognize that 36 can indeed be expressed as a power of 6: \[ 36 = 6^2 \] Now, the problem simplifies to solving for the exponent: \[ 6^x = 6^2 \] Since the bases are the same, we can equate the exponents directly: \[ x = 2 \] Therefore, \( \log_6 36 = 2 \).
Expressing numbers as powers helps us match the form needed to solve logarithmic problems quickly and accurately.
Solving Logarithmic Equations
To solve a logarithmic equation, we first need to understand what the equation is asking. Let's take the equation \( \log_6 36 \) again. It is asking for an exponent: 'What power of 6 gives us 36?'
Follow these steps:
  • Step 1: Recall that \( \log_b a \) means finding the exponent \( x \) such that \( b^x = a \).
  • Step 2: Express the number (inside the log) as a power of the base, if possible. Here, \( 36 = 6^2 \).
  • Step 3: Set up the equation \( b^x = b^2 \). Since the bases are the same (both are 6), we can directly equate the exponents: \[ x = 2 \]
  • Step 4: Write down the final answer: \( \log_6 36 = 2 \).
The key steps involve recognizing patterns in exponents and effectively using the properties of logarithms to simplify the equations. Remember that these principles can be applied to solve a wide array of logarithmic problems.

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Most popular questions from this chapter

In the following exercises, solve for \(x\), giving an exact answer as well as an approximation to three decimal places. \(6^{x}=91\)

In the following exercises, solve for \(x\). \(\log _{3} x+\log _{3} x=2\)

In the following exercises, solve for \(x\). \(\log _{5}(x+1)+\log _{5}(x-5)=\log _{5} 7\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{3} 36-\log _{3} 4\)

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible. \(\log _{2} 80-\log _{2} 5\)
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