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Chapter 4: Problem 33

Evaluate the expression without using a calculator.\(\log _{7} 7\)

Short Answer

Expert verified
The evaluated expression \(\log _{7} 7\) is 1.

Step by step solution

01

Understand the call of the question

The key to solving this exercise is understanding that the expression \(\log_{7}7\) represents the power to which the base number (7) must be raised to produce the number given within the log (also 7).
02

Apply Logarithmic Laws

When we apply the laws of logarithms it indicates that any value of 'a' where \(a > 0\), the result of \(\log_{a}a\) is equal to 1. Because in this case, you're trying to find the power to which the base (7) must be raised to result in the number 7 itself.
03

Conclude your answer

Therefore, \(\log_{7}7\) equals 1. This is because 7 to the power of 1 gives you 7, satisfying the conditions of a logarithmic expression.

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Key Concepts

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

Logarithmic Laws
Logarithmic laws are essential rules that simplify the manipulation and evaluation of logarithmic expressions. These laws are much like the laws of exponents, helping us understand and solve problems involving logarithms easily.
The most crucial law involved in this exercise is the identity law. The identity law states that for any base "a" where "a" is greater than zero, the logarithm of "a" to its own base is always 1. Mathematically, this is expressed as \(\log_{a}a = 1\).
  • The reason behind this is that the logarithm asks the question: "To what power must we raise 'a' to obtain 'a'?" The answer is clearly 1, since any number raised to the first power is itself.
This law becomes a handy shortcut in evaluating logarithmic expressions without needing complex calculations. Understanding this fundamental relationship makes solving logarithmic problems more intuitive.
Logarithmic Expressions
A logarithmic expression is a mathematical expression involving a logarithm. It's a way of expressing how many times you need to multiply a base number to get another number. In notation, it's written as \(\log_{a}b\), where "a" is the base and "b" is the number you want to achieve through repeated multiplication of "a".
  • The expression \(\log_{a}b\) essentially asks: "What exponent do we need for base 'a' to get 'b'?"
  • So, if \(b = a^x\), then \(\log_{a}b = x\).
It's important to become familiar with how to read and understand these expressions.
This helps in decoding the mathematical implications of what the logarithm is asking. The expression \(\log_{7}7\) illustrates this clearly, as it asks how many times we must multiply 7 by itself to get 7, which, unsurprisingly, is once.
Exponents
Exponents are a way to express repeated multiplication of a number by itself. If we have a number "a" raised to the power of "x", it means we're multiplying "a" by itself "x" times. This concept is fundamental in understanding both exponents and logarithms.
  • For instance, \(a^1 = a\), illustrating that any number raised to the power of 1 is the number itself.
  • This principle is pivotal to understanding the identity law of logarithms, such as \(\log_{a}a\).
The close relationship between exponents and logarithms lies in their inverse nature. While exponents represent multiplication, logarithms are about finding the necessary power from a given multiplication.
This interconnectedness is key to grasp, making it easier to transition between and solve problems involving both exponents and logarithms.

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Most popular questions from this chapter

Super Bowl Ad Cost The table shows the costs \(C\) (in millions of dollars) of a 30 -second TV ad during the Super Bowl for several years from 1987 to \(2006 .\) (Source: TNS Media Intelligence)$$ \begin{array}{|c|c|} \hline \text { Year } & \text { Cost } \\ \hline 1987 & 0.6 \\ \hline 1992 & 0.9 \\ \hline 1997 & 1.2 \\ \hline 2002 & 2.2 \\ \hline 2006 & 2.5 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1987 .\) (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use a graphing utility to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to predict the costs of a 30 -second ad during the Super Bowl in 2009 and in 2010 .

Solve for \(y\) in terms of \(x\).\(\ln y=2 \ln x+\ln (x-3)\)

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(\frac{119}{e^{6 x}-14}=7\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2\)

Aged Population The table shows the projected U.S. populations \(P\) (in thousands) of people who are 85 years old or older for several years from 2010 to \(2050 . \quad\) (Source: U.S. Census Bureau)$$ \begin{array}{|c|c|} \hline \text { Year } & 85 \text { years and older } \\ \hline 2010 & 6123 \\ \hline 2015 & 6822 \\ \hline 2020 & 7269 \\ \hline 2025 & 8011 \\ \hline 2030 & 9603 \\ \hline 2035 & 12,430 \\ \hline 2040 & 15,409 \\ \hline 2045 & 18,498 \\ \hline 2050 & 20,861 \\ \hline \end{array} $$(a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=10\) corresponding to 2010 . (b) Use the regression feature of a graphing utility to find an exponential model for the data. Use the Inverse Property \(b=e^{\ln b}\) to rewrite the model as an exponential model in base \(e\). (c) Use a graphing utility to graph the exponential model in base \(e\). (d) Use the exponential model in base \(e\) to estimate the populations of people who are 85 years old or older in 2022 and in 2042 .
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