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Chapter 12: Problem 8

Write in logarithmic form. \(\left(\frac{1}{6}\right)^{-3}=216\)

Short Answer

Expert verified
\text{log}_{\frac{1}{6}} (216) = -3.

Step by step solution

01

- Understand the exponential equation

Given the equation \(\frac{1}{6}^{-3} = 216\), recognize that this can be rewritten as \(\frac{1}{6}^{-3} = 216\).
02

- Apply the definition of a logarithm

Recall that the logarithm \(\text{log}_{b}(a) = c\) is the inverse operation of the exponential form \(b^c = a\).
03

- Rearrange the given equation

In the given equation, \(b = \frac{1}{6}\), \(a = 216\), and \(c = -3\). Therefore, it can be written in logarithmic form as \(\text{log}_{\frac{1}{6}} (216) = -3\).

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Key Concepts

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

Exponential Equations
An exponential equation is one where the variable appears in the exponent. In our example \(\frac{1}{6}^{-3} = 216\), the base \(\frac{1}{6}\) is raised to the power of \(-3\). Exponential equations often solve for unknown values based on their powers.
For example:
  • \(2^3 = 8\)
  • \(10^2 = 100\)
To solve these types of equations, we use inverse operations and logarithms.

Understanding exponential equations is the first step to grasping more complex logarithmic forms. Recognizing the base and its corresponding power within these equations helps transition to using logarithmic functions.
Inverse Operations
Inverse operations are pairs of mathematical operations that undo each other. For example, addition and subtraction are inverse operations and multiplication and division are inverses as well.
In the context of exponential equations and logarithms, the operations are inverses of each other.

For example:
  • If \(a^x = b\), then the logarithmic form is \(\text{log}_a(b) = x\).
    This means finding \(x\) in an exponential equation can be tackled by switching to its logarithmic form.
    • Multiplying asks:

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Most popular questions from this chapter

Solve each equation. \(\log _{5} x=-3\)

Solve each equation. \(\log _{x} 125=-3\)

Solve each equation. \(\log _{4} x=0\)

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{5} \frac{8}{3}$$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{2 x}=4 $$
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