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Chapter 3: Problem 106

evaluate each expression without using a calculator. $$\log _{5}\left(\log _{2} 32\right)$$

Short Answer

Expert verified
The evaluated expression is 1.

Step by step solution

01

Evaluate the Inner Logarithmic Expression

The first task is to evaluate the inner logarithmic expression \(\log _{2} 32\). Remember that \(\log _{b} a = c\) is equivalent to \(b^c = a\). Therefore, we are looking for a power to which we need to raise 2 to get 32. It is known that \(2^5 = 32\), so \(\log _{2} 32 = 5\).
02

Evaluate the Outer Logarithmic Expression

Now, we substitute the evaluated value from Step 1 into the outer logarithmic expression. Which gives us \(\log _{5} 5\). Again, using the fact that \(\log _{b} a = c\) is equivalent to \(b^c = a\), we can see that the power to which we need to raise 5 to get 5 is 1. Therefore, \(\log _{5} 5 = 1\).

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Key Concepts

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

Understanding Base Change
Logarithms allow us to represent exponential relationships, and changing the base can often make calculations simpler or more insightful. The base is the number we are raising to a power to get another number. For example, in \(\log_2 32\), the base is 2.
Changing the base might sound tricky, but there is a useful formula called the change of base formula:
  • \(\log_b a = \frac{\log_k a}{\log_k b}\)
This lets us compute logarithms with a different base using a base that is easier for us to work with, such as 10 or e (the natural logarithm).
In the given exercise, we didn't change the base, but it's important to know how to for more challenging problems or when using a calculator that only supports certain base logs.
Properties of Exponents and Logs
Exponents and logarithms are closely related. Logarithms are essentially the inverse operations of exponentials. This means they "undo" each other.
Here are some important properties:
  • If \(b^c = a\), then \(\log_b a = c\).
  • Logarithms of the same base let us simplify expressions: \(\log_b b = 1\)
  • Exponent rules apply: \(b^{m+n} = b^m \cdot b^n\).
These principles help break down complex logarithms into simpler forms. When evaluating \(\log_2 32\), we used the fact that 2 raised to the fifth power equals 32, illustrating why understanding these relationships is powerful.
Evaluating Expressions
Evaluation of logarithmic expressions involves breaking down the expression into manageable parts. Let's look at how this works with our initial problem.
We started with \(\log_{5}(\log_{2} 32)\). By solving from the inside out:
  • First, compute \(\log_{2} 32\). Since \(2^5 = 32\), \(\log_{2} 32 = 5\).
  • Substitute back into the outer expression, which becomes \(\log_{5} 5\).
  • Recognize that \(\log_{b} b = 1\), so \(\log_{5} 5 = 1\).
This approach鈥攕olving the innermost expression first鈥攕implifies the process tremendously and makes logarithmic evaluations straightforward. Practicing these steps will improve your problem-solving skills in math.

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Most popular questions from this chapter

If 4000 dollars is deposited into an account paying \(3 \%\) interest compounded annually and at the same time 2000 dollars is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance? Round to the nearest year.

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 .\) Use \(a\) graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65

Describe the following property using words: \(\log _{b} b^{x}=x\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I cannot simplify the expression \(b^{m}+b^{n}\) by adding exponents, there is no property for the logarithm of a sum.

By 2019 , nearly 1 dollar out of every 5 dollars spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product (GDP) going toward health care from 2007 through 2014 , with a projection for 2019.(GRAPH CAN'T COPY). The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2009 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.5 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.
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