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Chapter 4: Problem 17

Express the equation in logarithmic form. (a) \(10^{4}=10,000\) (b) \(5^{-2}=\frac{1}{25}\)

Short Answer

Expert verified
(a) \(\log_{10}{10,000} = 4\); (b) \(\log_{5}{\frac{1}{25}} = -2\).

Step by step solution

01

Understanding Exponential to Logarithmic Form

To convert an exponential form like \(a^b = c\) to logarithmic form, we use the relationship: \(\log_a{c} = b\). This means that 'b' is the exponent to which the base 'a' must be raised to get the result 'c'.
02

Convert Part (a)

In the equation \(10^4 = 10,000\), the base \(a = 10\), the exponent \(b = 4\), and the result \(c = 10,000\). Thus, the logarithmic form is \(\log_{10}{10,000} = 4\).
03

Convert Part (b)

For the equation \(5^{-2} = \frac{1}{25}\), the base \(a = 5\), the exponent \(b = -2\), and the result \(c = \frac{1}{25}\). Hence, the logarithmic form is \(\log_{5}{\frac{1}{25}} = -2\).

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Key Concepts

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

Understanding Exponential Equations
Exponential equations are mathematical expressions where a constant base is raised to a variable exponent. This can be seen in forms like \(a^b = c\). In such equations:
  • \(a\) is called the "base,"
  • \(b\) is the "exponent," or "power,"
  • \(c\) is the "result," or "product."
The beauty of exponential equations is their ability to express very large or very small numbers efficiently. For example, \(10^4\) simplifies to 10,000. Exponents can be positive, negative, or even fractions. Negative exponents, like in \(5^{-2} = \frac{1}{25}\), indicate division or reciprocals. Think of exponents as a process of repeated multiplication for positive exponents, or division for negative ones.
The Role of Logarithms
Logarithms serve as the inverse operation to exponentiation. If you understand a little bit about functions, you will know that inverse operations "undo" each other. If we know that \(a^b = c\), then the logarithm helps us find the exponent, \(b\). Logarithms answer the question: "To what power must the base be raised, to get a certain number?"
  • The base of the logarithm is the same as the base of the exponent.
  • The logarithmic expression is written as \(\log_a{c} = b\).
Using this relationship makes it far easier to solve equations involving exponentials and understand exponentiation in reverse. This reversibility means logs can turn multiplication processes into addition, division into subtraction, which are simpler operations.
Converting Between Exponential and Logarithmic Forms
Conversion between exponential and logarithmic forms relies on understanding their fundamental relationship. To express an equation like \(a^b = c\) in logarithmic form, we convert it to \(\log_a{c} = b\). This step is particularly useful when you solve equations where the exponent is unknown and you want to rewrite it to make this unknown the subject.

Consider the given equations from the textbook exercise:
  • \(10^4 = 10,000\) converts to \(\log_{10}{10,000} = 4\).
  • \(5^{-2} = \frac{1}{25}\) converts to \(\log_{5}{\frac{1}{25}} = -2\).
By rearranging the exponential equation into this form, the logarithm provides clarity and highlights the hidden exponent. It's a powerful tool for anyone dealing with complex calculations involving scales, especially in fields like science and engineering.

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Most popular questions from this chapter

The count in a culture of bacteria was 400 after 2 hours and \(25,600\) after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (d) Find the number of bacteria after 4.5 hours. (e) After how many hours will the number of bacteria reach \(50.000 ?\)

Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. $$7^{x / 2}=5^{1-x}$$

Compare the graphs of the power function \(f\) and exponential function \(g\) by evaluating both of them for \(x=0,1,2,3,4,6,8,\) and 10 Then draw the graphs of \(f\) and \(g\) on the same set of axes. \(f(x)=x^{4} ; \quad g(x)=4^{x}\)

Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. $$2^{3 x+1}=3^{x-2}$$

The hydrogen ion concentration of a sample of each substance is given. Calculate the \(\mathrm{pH}\) of the substance. (a) Lemon juice: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-3} \mathrm{M}\) (b) Tomato juice: \(\left[\mathrm{H}^{+}\right]=3.2 \times 10^{-4} \mathrm{M}\) (c) Seawater: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-9} \mathrm{M}\)
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