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Chapter 11: Problem 31

Evaluate without a calculator, or say if the expression is undefined. $$ 10^{\log 1} $$

Short Answer

Expert verified
Answer: The value of the expression $$10^{\log 1}$$ is 1.

Step by step solution

01

Recall the property for the logarithm of 1

Recall the property of logarithms stating that the base b logarithm of 1 is always 0, regardless of the value of b: $$\log_b 1 = 0$$ In this case, our base is 10, so we know that $$\log_{10} 1 = 0$$
02

Substitute the logarithm value into the expression

Substitute the value of the logarithm into the expression: $$10^{\log 1} = 10^0$$
03

Evaluate the expression

Recall the property of exponents that states any number raised to the power of 0 is 1: $$a^0 = 1$$ Using this property, the expression simplifies to: $$10^0 = 1$$ So, the expression $$10^{\log 1}$$ is equal to $$1$$.

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Key Concepts

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

Properties of Logarithms
Logarithms are essential tools in mathematics that help us solve equations involving exponents. One fundamental property of logarithms is that the logarithm of 1 is always 0, regardless of the base. For any base \( b \), we observe this using the formula \( \log_b 1 = 0 \). This property stems from the understanding that any number raised to the power of 0 equals 1, which is the inverse process of logarithms.

By knowing that \( \log_{10} 1 = 0 \), students can simplify expressions involving log and exponent operations efficiently. This foundational knowledge assists in evaluating expressions like \( 10^{\log 1} \) quickly, making log-based problems more approachable.
  • Learn the core log rule: \( \log_b 1 = 0 \).
  • Practice switching between exponential and log forms to enhance problem-solving skills.
Properties of Exponents
Exponents describe how many times a number, known as the base, is multiplied by itself. One key property is that any non-zero number raised to the power of zero equals one, expressed as \( a^0 = 1 \). This rule is intuitive if you think of decreasing exponents sequentially down to zero.

For instance, consider consecutive powers like \( a^3, a^2, a^1 \). Each step down is the result of dividing the previous power by \( a \). By continuing this pattern, we find \( a^0 = 1 \) supports the logical consistency.
  • Understand that \( a^0 = 1 \) for any non-zero \( a \).
  • Remembering this can simplify seemingly complex problems.
Evaluating Expressions
Evaluating expressions involves reducing mathematical problems into simpler, solvable components using known properties. In the context of exponential and logarithmic expressions, start by applying the appropriate logarithmic or exponential property.

In our example, to evaluate \( 10^{\log 1} \), identify that \( \log_{10} 1 = 0 \). Substitute this into the expression to simplify it to \( 10^0 \). Then, use the exponential property \( a^0 = 1 \) to find the solution.
  • Break down expressions into understandable steps.
  • Use initial log or exponent insights to simplify.

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Most popular questions from this chapter

In \(1936,\) researchers at the Wright-Patterson Air Force Base found that each new aircraft took less time to produce than the one before, owing to what is now known as the learning curve effect. Specifically, researchers found that \(^{16}\) $$ \begin{aligned} f(n) &=t_{0} n^{\log _{2} k} \\ \text { where } f(n) &=\text { Time required to produce } n^{\text {th }} \text { unit } \end{aligned} $$ \(t_{0}=\) Time required to produce the first unit. The value of \(k\) will vary from industry to industry, but for a particular industry it is often a constant. (a) Suppose for a particular industry, the first unit takes 10,000 hours of labor to produce, so \(n_{0}=\) \(10,000,\) and that \(k=0.8\). The learning-curve effect states that the additional hours of labor goes down by a fixed percentage each time production doubles. Show that this is the case by completing the table. By what percent does the required time drop when production is doubled? $$ \begin{array}{c|c|c|c|c|c} \hline n & 1 & 2 & 4 & 8 & 16 \\ \hline f(n) & & & & & \\ \hline \end{array} $$ (b) What kind of function (exponential, power, logarithmic, other) is \(f(n) ?\) Discuss. (c) Suppose a different industry has a lower value of \(t_{0}\), say, \(t_{0}=7000\), but a higher value of \(b\), say, \(b=0.9 .\) Explain what this tells you about the difference in the learning-curve effect between these two industries. (d) What would it mean for \(k>1\) in terms of production time? Explain why this is unlikely to be the case.

Assume \(a\) and \(b\) are positive constants. Imagine solving for \(x\) (but do not actually do so). Will your answer involve logarithms? Explain how you can tell. $$ 10^{x}=a $$

Solve the equations, first approximately, as in Example \(1,\) by filling in the given table, and th猫n to four decimal places by using logarithms. $$ \begin{aligned} &\text { Table } 11.5 \text { Solve }\\\ &10^{x}=0.03\\\ &\begin{array}{c|c|c|c|c} \hline x & -1.6 & -1.5 & -1.4 & -1.3 \\ \hline 10^{x} & & & & \\ \hline \end{array} \end{aligned} $$

If possible, use logarithm properties to rewrite the expressions in terms of \(u, v, w\) given that $$u=\log x, v=\log y, w=\log z$$ Your answers should not involve logs. $$ \log x^{2} y^{3} \sqrt{z} $$

If possible, use logarithm properties to rewrite the expressions in terms of \(u, v, w\) given that $$u=\log x, v=\log y, w=\log z$$ Your answers should not involve logs. $$ \log x y $$
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