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Chapter 4: Problem 30

Write the equation in logarithmic form. $$ \left(\frac{1}{2}\right)^{-5}=32 $$

Short Answer

Expert verified
\(\text{log}_{\frac{1}{2}}(32) = -5\)

Step by step solution

01

Identify the Components

Recognize the components of the exponential equation \(\frac{1}{2}^{-5} = 32\). Here, \(\frac{1}{2}\) is the base, \(-5\) is the exponent, and \(32\) is the result.
02

Understand the Logarithmic Form

Recall that the logarithmic form of \(b^e = a\) is \(\text{log}_b(a) = e\), where \(b\) is the base, \(e\) is the exponent, and \(a\) is the result.
03

Apply the Logarithmic Form

Use the logarithmic form to convert the given exponential equation \(\frac{1}{2}^{-5} = 32\) into logarithmic form: \(\text{log}_{\frac{1}{2}}(32) = -5\).

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Key Concepts

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

Exponential Equation
An exponential equation is a type of equation in which a variable appears in the exponent. For example, the equation \(\frac{1}{2}^{-5} = 32\) is an exponential equation because the variable term involves an exponent. Understanding this type of equation is crucial for solving many mathematical problems.
In general terms, an exponential equation looks like this: \(b^e = a\), where:
  • \textbf{b} is the base
  • \textbf{e} is the exponent
  • \textbf{a} is the result produced
With these equations, our goal is often to find the exponent or to express the equation in a different form, such as a logarithmic form. An exponential equation has unique properties, making it significantly different from simple linear or polynomial equations.
Logarithms
Logarithms are the inverse functions of exponentials. While an exponential function involves raising a base to a power to get a result, a logarithm does the reverse鈥攊t determines what power you need to raise the base to get that result. The logarithmic form of the exponential equation \(\frac{1}{2}^{-5} = 32\) is \(\text{log}_{\frac{1}{2}}(32) = -5\).
Here,
  • \textbf{Base}: The number we are raising to a power (in this case, \(\frac{1}{2}\))
  • \textbf{Result}: The number obtained from the exponential expression (in this case, \(32\))
  • \textbf{Exponent}: The power to which the base is raised (in this case, \(-5\))
Essentially, logarithms provide another way to represent exponential relationships, and they are particularly useful when dealing with very large or small numbers, as they simplify multiplication and division into addition and subtraction.
Base and Exponent
To fully understand exponential equations and logarithms, it is essential to understand the concepts of base and exponent. The base is the constant value that is being multiplied. It is the number that serves as the starting point. In the equation \(\frac{1}{2}^{-5} = 32\), the base is \( \frac{1}{2} \).
The exponent is the power to which the base is raised, determining how many times the base is multiplied by itself. In the given equation, the exponent is \(-5\). Exponents can be positive, negative, or zero:
  • \textbf{Positive Exponent}: Indicates repeated multiplication, such as \(2^3 = 8\)
  • \textbf{Negative Exponent}: Indicates repeated division or taking the reciprocal, like \(2^{-3} = \frac{1}{8}\)
  • \textbf{Zero Exponent}: Any non-zero number raised to the power of zero is 1, like \(2^0 = 1\)
By mastering the role of the base and exponent in equations, you'll be better equipped to solve and transform exponential equations into logarithmic form.

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Most popular questions from this chapter

Use the change-of-base formula to write \(\left(\log _{2} 5\right)\left(\log _{5} 9\right)\) as a single logarithm.

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log \left(p^{2}+6 p\right)=\log 7\)

Solve for the indicated variable. \(\log E-12.2=1.44 M\) for \(E\) (used in geology)

Determine if the given value of \(x\) is a solution to the logarithmic equation. \(\log _{2}(x-31)=5-\log _{2} x\) a. \(x=16\) b. \(x=32\) c. \(x=-1\)

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{2} 15 $$
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