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Chapter 9: Problem 15

Simplify. $$ \log _{7} 49 $$

Short Answer

Expert verified
\( \log _{7} 49 = 2 \)

Step by step solution

01

Understand the Problem

The task is to simplify the logarithmic expression \(\log _{7} 49\). This expression asks the question: 'To what power must 7 be raised to obtain 49?'
02

Recognize the Relationship

Recall that \( 49 = 7^2 \). This is because 7 multiplied by itself (7 \times 7) equals 49.
03

Apply the Logarithmic Rule

Using the relationship from Step 2, apply the logarithmic rule: \( \log _{b} (b^k) = k \). In this case, \( b = 7 \) and \( k = 2 \).
04

Simplify the Expression

Therefore, \( \log _{7} 49 = \log _{7} (7^2) = 2 \), because 7 must be raised to the power of 2 to get 49.

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Key Concepts

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

Logarithmic Rules
Understanding logarithmic rules is key to simplifying logarithmic expressions. A logarithm tells us the exponent needed to raise a base to get a particular number. For instance, in the expression \(\text{log}_b(a)\), 'b' is the base and 'a' is the number you're finding the logarithm of.

An important rule to remember is the power rule: \(\text{log}_b(b^k) = k\). This means when you take the logarithm of a base raised to a power, the result is just the exponent. For example:

    \(\text{log}_2(2^3) = 3\), because 2 raised to the power of 3 gives us 8.

    When simplifying, always check for this rule as it allows you to reduce complex expressions to simpler forms.
Exponents
Exponents represent repeated multiplication. For example, \({2^3} = 2 \times 2 \times 2\). Here, 2 is the base, and 3 is the exponent. Exponents radically simplify notation, especially for large numbers.

Be familiar with these basic exponent rules:
  • \(b^m \times b^n = b^{m+n}\)
  • \(b^m / b^n = b^{m-n}\)
  • \((b^m)^n = b^{mn}\)

Understanding these rules can make working with exponents easier. For example, knowing that \(49 = 7^2\) helps you connect it to logarithms quickly since logarithms are essentially the inverse operation of exponentiation.
Base and Power Relationship
In logarithms, understanding the base and power relationship is crucial. A logarithm's fundamental question is, 'How many times do we multiply the base to get the number?'

Consider \(log_7(49)\): We're asking, 'What power should 7 be raised to, to get 49?' Since \(7^2 = 49\), the answer is 2.
The base must be raised to the power to achieve the number.

To generalize: In any expression \(log_b(a)\), you find 'c' if \(b^c = a\). For another way of visualizing:
  • \text{log}_2(8) = 3: This asks 'What power raises 2 to get 8?' Since \(2^3 = 8\).
  • \text{log}_5(125) = 3: Here we're asking 'What power raises 5 to get 125?' Because \(5^3 = 125 \).

Understanding these relationships can demystify many logarithmic problems and make solving them more intuitive.

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Most popular questions from this chapter

In 2016 , the population of the United States was 324 million and the exponential growth rate was \(0.73 \%\) per year. Data: U.S. Census Bureau a) Find the exponential growth function. b) Estimate the U.S. population in 2025 c) When will the U.S. population reach 400 million?

The average price paid for a Super Bowl ticket has increased exponentially from 12 dollars in 1967 to 2670 dollars in 2015 Data: DallasNews.com; seatgeek.com a) Find the exponential growth rate \(k,\) and write an equation for an exponential function that can be used to predict the average price paid for a Super Bowl ticket \(t\) years after 1967 b) Estimate the year in which the average price paid for a Super Bowl ticket will reach 5000 dollars.

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

Is there a one-to-one relationship between items in a store and the price of each of those items? Why or why not?

In 2016 , the world population was 7.4 billion and the exponential growth rate was \(1.13 \%\) per year. Data: U.S. Census Bureau a) Find the exponential growth function. b) Estimate the world population in 2025 c) When will the world population reach 10 billion?
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